At the current course, the seminars began to consist not in solving problems, but in reports on various topics. I think it will be right to leave them here in a more or less popular form.
The word "soliton" comes from the English solitary wave and means exactly a solitary wave (or, in the language of physics, some excitation).
Soliton near Molokai Island (Hawaiian archipelago)
A tsunami is also a soliton, but much larger. Solitude does not mean that there will be only one wave in the whole world. Solitons are sometimes found in groups, as near Burma.
Solitons in the Andaman Sea washing the shores of Burma, Bengal and Thailand.
In a mathematical sense, a soliton is a solution to a non-linear partial differential equation. This means the following. To solve linear equations that are ordinary from school, that differential humanity has already been able to do it for a long time. But as soon as a square, a cube, or an even more cunning dependence arises in a differential equation from an unknown quantity, the mathematical apparatus that has been developed over the centuries fails - a person has not yet learned how to solve them and solutions are most often guessed or selected from various considerations. But they describe Nature. So non-linear dependencies give rise to almost all phenomena that enchant the eye, and allow life to exist too. The rainbow in its mathematical depth is described by the Airy function (although speaking surname for a scientist whose research tells about the rainbow?)
The contractions of the human heart are a typical example of biochemical processes called autocatalytic - those that maintain their own existence. All linear dependencies and direct proportions, although simple for analysis, are boring: nothing changes in them, because the straight line remains the same at the origin and goes to infinity. More complex functions have special points: minima, maxima, faults, etc., which, once in the equation, create countless variations for the development of systems.
Functions, objects or phenomena called solitons have two important properties: they are stable over time and they retain their shape. Of course, in life, no one and nothing will satisfy them indefinitely, so you need to compare with similar phenomena. Returning to the sea surface, ripples on its surface appear and disappear in a fraction of a second, large waves raised by the wind take off and scatter with spray. But the tsunami moves like a blank wall for hundreds of kilometers without losing noticeably in wave height and strength.
There are several types of equations leading to solitons. First of all, this is the Sturm-Liouville problem
IN quantum theory this equation is known as the non-linear Schrödinger equation (Schrödinger) if the function has an arbitrary form. In this notation, the number is called its own. It is so special that it is also found when solving a problem, because not every value of it can give a solution. The role of eigenvalues in physics is very great. For example, energy is an eigenvalue in quantum mechanics, transitions between different coordinate systems also cannot do without them. If you require that a parameter change t did not change their own numbers (and t can be time, for example, or some external influence on the physical system), then we arrive at the Korteweg-de Vries equation:
There are other equations, but now they are not so important.
In optics, the phenomenon of dispersion plays a fundamental role - the dependence of the frequency of a wave on its length, or rather the so-called wave number:
In the simplest case, it can be linear (, where is the speed of light). In life, we often get the square of the wave number, or even something more tricky. In practice, dispersion limits the bandwidth of the fiber that those words just ran to your ISP from the WordPress servers. But it also allows you to pass through one optical fiber not one beam, but several. And in terms of optics, the above equations consider the simplest cases of dispersion.
Solitons can be classified in different ways. For example, solitons that appear as some kind of mathematical abstraction in systems without friction and other energy losses are called conservative. If we consider the same tsunami for a not very long time (and it should be more useful for health), then it will be a conservative soliton. Other solitons exist only due to the flows of matter and energy. They are usually called autosolitons, and further we will talk about autosolitons.
In optics, they also talk about temporal and spatial solitons. From the name it becomes clear whether we will observe a soliton as a kind of wave in space, or whether it will be a surge in time. Temporal ones arise due to the balancing of nonlinear effects by diffraction - the deviation of rays from rectilinear propagation. For example, they shone a laser into glass (optical fiber), and inside the laser beam the refractive index began to depend on the power of the laser. Spatial solitons arise due to the balancing of nonlinearities by dispersion.
Fundamental soliton
As already mentioned, broadband (that is, the ability to transmit many frequencies, and hence useful information) of fiber-optic communication lines is limited by non-linear effects and dispersion, which change the amplitude of the signals and their frequency. But on the other hand, the same nonlinearity and dispersion can lead to the creation of solitons that retain their shape and other parameters much longer than anything else. The natural conclusion from this is the desire to use the soliton itself as an information signal (there is a flash-soliton at the end of the fiber - a one was transmitted, no - a zero was transmitted).
An example with a laser that changes the refractive index inside an optical fiber as it propagates is quite vital, especially if you “shove” a pulse of several watts into a fiber thinner than a human hair. By comparison, a lot or not, a typical 9W energy-saving light bulb illuminates a desk, but is about the size of a palm. In general, we will not deviate far from reality by assuming that the dependence of the refractive index on the pulse power inside the fiber will look like this:
After physical reflections and mathematical transformations of varying complexity per amplitude electric field inside the fiber, one can obtain an equation of the form
where and is the coordinate along the propagation of the beam and transverse to it. The coefficient plays an important role. It defines the relationship between dispersion and non-linearity. If it is very small, then the last term in the formula can be thrown out due to the weakness of the nonlinearities. If it is very large, then the nonlinearities, having crushed the diffraction, will single-handedly determine the features of the signal propagation. So far, attempts have been made to solve this equation only for integer values of . So when the result is especially simple:
.
The hyperbolic secant function, although it is called long, looks like an ordinary bell
Intensity distribution in the cross section of a laser beam in the form of a fundamental soliton.
It is this solution that is called the fundamental soliton. The imaginary exponent determines the propagation of the soliton along the fiber axis. In practice, this all means that if we shine on the wall, we would see a bright spot in the center, the intensity of which would quickly decrease at the edges.
The fundamental soliton, like all solitons that arise with the use of lasers, has certain features. First, if the laser power is insufficient, it will not appear. Secondly, even if somewhere the locksmith overbends the fiber, drops oil on it or does some other dirty trick, the soliton, passing through the damaged area, will be indignant (in the physical and figurative sense), but will quickly return to its original parameters. People and other living beings also fall under the definition of an autosoliton, and this ability to return to a calm state is very important in life 😉
The energy flows inside the fundamental soliton look like this:
Direction of energy flows inside the fundamental soliton.
Here, the circle separates the areas with different flow directions, and the arrows indicate the direction.
In practice, several solitons can be obtained if the laser has several generation channels parallel to its axis. Then the interaction of solitons will be determined by the degree of overlap of their "skirts". If the energy dissipation is not very large, we can assume that the energy fluxes inside each soliton are conserved in time. Then the solitons start spinning and sticking together. The following figure shows a simulation of the collision of two triplets of solitons.
Simulation of the collision of solitons. Amplitudes are shown on a gray background (as a relief), and phase distribution is shown on black.
Groups of solitons meet, cling, and forming a Z-like structure begin to rotate. Even more interesting results can be obtained by breaking the symmetry. If you place laser solitons in a checkerboard pattern and discard one, the structure will begin to rotate.
Symmetry breaking in a group of solitons leads to the rotation of the center of inertia of the structure in the direction of the arrow in Fig. to the right and rotation around the instantaneous position of the center of inertia
There will be two rotations. The center of inertia will turn counterclockwise, and the structure itself will rotate around its position at each moment of time. Moreover, the periods of rotation will be equal, for example, like that of the Earth and the Moon, which is turned to our planet with only one side.
Experiments
Such unusual properties of solitons have attracted attention and made us think about practical applications for about 40 years. We can immediately say that solitons can be used to compress pulses. To date, it is possible to obtain a pulse duration of up to 6 femtoseconds in this way (sec or take one millionth of a second twice and divide the result by a thousand). Of particular interest are soliton communication lines, the development of which has been going on for quite a long time. So Hasegawa proposed the following scheme back in 1983.
Soliton communication line.
The communication line is formed from sections about 50 km long. The total length of the line was 600 km. Each section consists of a receiver with a laser transmitting an amplified signal to the next waveguide, which made it possible to achieve a speed of 160 Gbit / s.
Presentation
Literature
- J. Lem. Introduction to the theory of solitons. Per. from English. M.: Mir, - 1983. -294 p.
- J. Whitham Linear and non-linear waves. - M.: Mir, 1977. - 624 p.
- I. R. Shen. Principles of nonlinear optics: Per. from English / Ed. S. A. Akhmanova. - M.: Nauka., 1989. - 560 p.
- S. A. Bulgakova, A. L. Dmitriev. Nonlinear-optical information processing devices// Textbook. - St. Petersburg: SPbGUITMO, 2009. - 56 p.
- Werner Alpers et. al. Observation of Internal Waves in the Andaman Sea by ERS SAR // Earthnet Online
- A. I. Latkin, A. V. Yakasov. Autosoliton regimes of pulse propagation in a fiber-optic communication line with nonlinear ring mirrors // Avtometriya, 4 (2004), v.40.
- N. N. Rozanov. World of laser solitons // Nature, 6 (2006). pp. 51-60.
- O. A. Tatarkina. Some aspects of designing soliton fiber-optic transmission systems // Basic Research, 1 (2006), pp. 83-84.
P.S. About diagrams in .
A person, even without a special physical or technical education the words "electron, proton, neutron, photon" are undoubtedly familiar. But the word “soliton”, which is consonant with them, is probably heard by many for the first time. This is not surprising: although what is denoted by this word has been known for more than a century and a half, proper attention has been paid to solitons only since the last third of the 20th century. Soliton phenomena turned out to be universal and were found in mathematics, hydromechanics, acoustics, radiophysics, astrophysics, biology, oceanography, and optical engineering. What is a soliton?
All of the above areas have one common feature: in them or in their individual sections, wave processes are studied, or, more simply, waves. In the most general sense, a wave is the propagation of a disturbance of some physical quantity characterizing the substance or field. This propagation usually occurs in some medium - water, air, solids Oh. And only electromagnetic waves can propagate in a vacuum. Everyone, no doubt, saw how spherical waves diverge from a stone thrown into the water, which “disturbed” the calm surface of the water. This is an example of the propagation of a "single" perturbation. Very often, the perturbation is an oscillatory process (in particular, periodic) in various forms - the swing of a pendulum, the vibration of a musical instrument string, the compression and expansion of a quartz plate under the action of an alternating current, vibrations in atoms and molecules. Waves - propagating oscillations - can have a different nature: waves on water, sound, electromagnetic (including light) waves. The difference in the physical mechanisms that implement the wave process entails different ways of its mathematical description. But waves of different origin also have some common properties, which are described using a universal mathematical apparatus. And this means that it is possible to study wave phenomena, abstracting from their physical nature.
In wave theory, this is usually done, considering such properties of waves as interference, diffraction, dispersion, scattering, reflection, and refraction. But in this case, one important circumstance takes place: such a unified approach is justified provided that the studied wave processes of different nature are linear. We will talk about what is meant by this a little later, but for now we will only note that only waves with not too large amplitude can be linear. If the wave amplitude is large, it becomes nonlinear, and this is directly related to the topic of our article - solitons.
Since we talk about waves all the time, it is not difficult to guess that solitons are also something from the field of waves. This is true: a very unusual formation is called a soliton - a "solitary" wave (solitary wave). The mechanism of its occurrence has long remained a mystery to researchers; it seemed that the nature of this phenomenon contradicted the well-known laws of the formation and propagation of waves. Clarity appeared relatively recently, and now they are studying solitons in crystals, magnetic materials, fiber light guides, in the atmosphere of the Earth and other planets, in galaxies and even in living organisms. It turned out that tsunamis, and nerve impulses, and dislocations in crystals (violations of the periodicity of their lattices) are all solitons! Soliton is truly "many-sided". By the way, this is the name of A. Filippov's excellent popular science book "The Many-Faced Soliton". We recommend it to the reader who is not afraid enough a large number mathematical formulas.
In order to understand the basic ideas associated with solitons, and at the same time to do practically without mathematics, we will have to talk first of all about the already mentioned nonlinearity and dispersion - the phenomena underlying the mechanism of soliton formation. But first, let's talk about how and when the soliton was discovered. He first appeared to man in the "guise" of a solitary wave on the water.
... It happened in 1834. John Scott Russell, a Scottish physicist and talented engineer-inventor, was invited to investigate the possibility of navigating steam ships along the canal connecting Edinburgh and Glasgow. At that time, transportation along the canal was carried out using small barges pulled by horses. In order to figure out how to convert barges when replacing horse-powered with steam, Russell began to observe barges of various shapes moving from different speeds. And in the course of these experiments, he suddenly encountered a completely unusual phenomenon. This is how he described it in his Report on the Waves:
“I was following the movement of a barge, which was quickly pulled along a narrow channel by a couple of horses, when the barge suddenly stopped. But the mass of water that the barge set in motion gathered near the bow of the ship in a state of frenzied motion, then unexpectedly left it behind, rolling forward at great speed and taking the form of a large single elevation - a rounded, smooth and well-defined water hill. It continued along the canal without changing its shape or slowing down in the slightest. I followed him on horseback, and when I overtook him, he was still rolling forward at about 8 or 9 miles per hour, retaining his original elevation profile, about thirty feet long and a foot to a foot and a half high. Its height gradually decreased, and after one or two miles of pursuit I lost it in the bends of the canal.
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Russell called the phenomenon he discovered "the solitary wave of translation." However, his message was met with skepticism by recognized authorities in the field of hydrodynamics - George Airy and George Stokes, who believed that waves moving over long distances cannot maintain their shape. For this they had every reason: they proceeded from the equations of hydrodynamics generally accepted at that time. The recognition of a "solitary" wave (which was called a soliton much later - in 1965) occurred during Russell's lifetime by the works of several mathematicians who showed that it can exist, and, in addition, Russell's experiments were repeated and confirmed. But the controversy around the soliton did not stop for a long time - the authority of Airy and Stokes was too great.
The Dutch scientist Diderik Johannes Korteweg and his student Gustav de Vries brought final clarity to the problem. In 1895, thirteen years after Russell's death, they found the exact equation, the wave solutions of which completely describe the ongoing processes. As a first approximation, this can be explained as follows. Korteweg–de Vries waves have a non-sinusoidal shape and become sinusoidal only when their amplitude is very small. With an increase in the wavelength, they take the form of humps that are far apart from each other, and at a very large wavelength, one hump remains, which corresponds to the "solitary" wave.
The Korteweg - de Vries equation (the so-called KdV equation) has played a very important role in our days, when physicists have realized its universality and the possibility of application to waves of various nature. The most remarkable thing is that it describes nonlinear waves, and now we should dwell on this concept in more detail.
In the theory of waves, the wave equation is of fundamental importance. Without presenting it here (this requires familiarity with higher mathematics), we only note that the desired function describing the wave and the quantities associated with it are contained in the first degree. Such equations are called linear. The wave equation, like any other, has a solution, that is, a mathematical expression, which, when substituted, turns into an identity. The solution to the wave equation is a linear harmonic (sinusoidal) wave. We emphasize once again that the term "linear" is used here not in a geometric sense (a sinusoid is not a straight line), but in the sense of using the first power of quantities in the wave equation.
Linear waves obey the principle of superposition (addition). This means that when several linear waves are superimposed, the shape of the resulting wave is determined by a simple addition of the original waves. This happens because each wave propagates in the medium independently of the others, there is no energy exchange or other interaction between them, they freely pass through one another. In other words, the principle of superposition means the independence of the waves, and that is why they can be added. Under normal conditions, this is true for sound, light and radio waves, as well as for waves that are considered in quantum theory. But for waves in a liquid, this is not always true: only waves of very small amplitude can be added. If we try to add the Korteweg-de Vries waves, then we will not get a wave at all that can exist: the equations of hydrodynamics are nonlinear.
It is important to emphasize here that the linearity property of acoustic and electromagnetic waves is observed, as already noted, under normal conditions, which mean, first of all, small wave amplitudes. But what does “small amplitudes” mean? The amplitude of sound waves determines the volume of sound, light waves determine the intensity of light, and radio waves determine the strength of the electromagnetic field. Broadcasting, television, telephones, computers, lighting fixtures, and many other devices operate in the same "normal environment", dealing with a variety of small amplitude waves. If the amplitude sharply increases, the waves lose their linearity and then new phenomena arise. In acoustics, shock waves propagating at supersonic speeds have long been known. Examples of shock waves are thunder during a thunderstorm, the sound of a gunshot and explosion, and even the clapping of a whip: its tip moves faster than sound. Nonlinear light waves are obtained using powerful pulsed lasers. The passage of such waves through various environments changes the properties of the media themselves; completely new phenomena are observed, which are the subject of study of nonlinear optics. For example, a light wave arises, the length of which is two times smaller, and the frequency, respectively, twice that of the incoming light (the second harmonic is generated). If, say, a powerful laser beam with a wavelength of λ 1 = 1.06 μm (infrared radiation, invisible to the eye) is directed to a nonlinear crystal, then green light with a wavelength of λ 2 = 0.53 μm appears at the output of the crystal in addition to infrared.
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This is how a nonlinear wave behaves on the water surface in the absence of dispersion. Its speed does not depend on the wavelength, but increases with increasing amplitude. The crest of the wave moves faster than the bottom, the front gets steeper and the wave breaks. But a solitary hump on the water can be represented as a sum of components with different wavelengths. If the medium has dispersion, long waves in it will run faster than short ones, leveling the steepness of the front. Under certain conditions, dispersion completely compensates for the effect of nonlinearity, and the wave will retain its original shape for a long time - a soliton is formed. |
If non-linear sound and light waves are formed only under special conditions, then hydrodynamics is non-linear by its very nature. And since hydrodynamics exhibits nonlinearity even in the simplest phenomena, for almost a century it has been developing in complete isolation from "linear" physics. It simply never occurred to anyone to look for anything similar to Russell's "solitary" wave in other wave phenomena. And only when new areas of physics were developed - nonlinear acoustics, radio physics and optics - the researchers remembered the Russell soliton and asked the question: can such a phenomenon be observed only in water? To do this, it was necessary to understand the general mechanism of soliton formation. The condition of nonlinearity turned out to be necessary, but not sufficient: something else was required from the medium so that a “solitary” wave could be born in it. And as a result of the research, it became clear that the missing condition was the presence of dispersion of the medium.
Let us briefly recall what it is. Dispersion is the dependence of the propagation velocity of the wave phase (the so-called phase velocity) on the frequency or, what is the same, the wavelength (see "Science and Life" No. 2, 2000, p. 42). According to the well-known Fourier theorem, a non-sinusoidal wave of any shape can be represented by a set of simple sinusoidal components with different frequencies (wavelengths), amplitudes and initial phases. These components, due to dispersion, propagate at different phase velocities, which leads to the "smearing" of the waveform as it propagates. But the soliton, which can also be represented as the sum of these components, as we already know, retains its shape when moving. Why? Recall that a soliton is a non-linear wave. And here lies the key to unlocking his "secret". It turns out that a soliton arises when the effect of nonlinearity, which makes the "hump" of the soliton steeper and tends to overturn it, is balanced by dispersion, which makes it flatter and tends to blur it. That is, a soliton appears "at the junction" of nonlinearity and dispersion, compensating each other.
Let's explain this with an example. Suppose that a hump formed on the surface of the water, which began to move. Let's see what happens if we do not take into account the dispersion. The speed of a nonlinear wave depends on the amplitude (linear waves do not have such a dependence). The top of the hump will move fastest of all, and at some next moment its front will become steeper. The steepness of the front increases, and in the course of time, the wave will "overturn". We see a similar overturning of the waves when we watch the surf on the seashore. Now let's see what the presence of dispersion leads to. The initial hump can be represented by the sum of sinusoidal components with different wavelengths. The long-wave components run at a higher speed than the short-wave ones, and, therefore, reduce the steepness of the leading edge, to a large extent leveling it (see "Science and Life" No. 8, 1992). At a certain shape and speed of the hump, a complete restoration of the original shape can occur, and then a soliton is formed.
One of the amazing properties of "solitary" waves is that they are a lot like particles. So, in a collision, two solitons do not pass through each other, like ordinary linear waves, but, as it were, repel each other like tennis balls.
Solitons of another type, called group solitons, can also appear on water, since their shape is very similar to groups of waves, which in reality are observed instead of an infinite sinusoidal wave and move with a group velocity. The group soliton closely resembles amplitude-modulated electromagnetic waves; its envelope is non-sinusoidal; it is described by a more complex function, the hyperbolic secant. The velocity of such a soliton does not depend on the amplitude, and in this respect it differs from KdV solitons. Under the envelope is usually no more than 14 - 20 waves. The middle - the highest - wave in the group is thus in the interval from the seventh to the tenth; hence the well-known expression "the ninth wave".
The scope of the article does not allow us to consider many other types of solitons, for example, solitons in solid crystalline bodies - the so-called dislocations (they resemble "holes" in crystal lattice and are also able to move), magnetic solitons related to them in ferromagnets (for example, in iron), soliton-like nerve impulses in living organisms, and many others. We confine ourselves to consideration of optical solitons, which in Lately attracted the attention of physicists by the possibility of their use in very promising optical communication lines.
An optical soliton is a typical group soliton. Its formation can be understood on the example of one of the nonlinear optical effects - the so-called self-induced transparency. This effect consists in the fact that a medium that absorbs light of low intensity, that is, opaque, suddenly becomes transparent when a powerful light pulse passes through it. To understand why this happens, let us recall what causes the absorption of light in matter.
A light quantum, interacting with an atom, gives it energy and transfers it to a higher energy level, that is, to an excited state. In this case, the photon disappears - the medium absorbs light. After all the atoms of the medium are excited, the absorption of light energy stops - the medium becomes transparent. But such a state cannot last long: the photons flying behind cause the atoms to return to their original state, emitting quanta of the same frequency. This is exactly what happens when a short light pulse of high power of the corresponding frequency is directed through such a medium. The leading edge of the pulse throws the atoms to the upper level, being partially absorbed and becoming weaker. The maximum of the pulse is absorbed to a lesser extent, and the trailing edge of the pulse stimulates the reverse transition from the excited level to the ground level. The atom emits a photon, its energy is returned to the impulse, which passes through the medium. In this case, the shape of the pulse turns out to correspond to a group soliton.
Quite recently, one of the American scientific journals published a publication on the ongoing development of the well-known company "Bell" (Bell Laboratories, USA, New Jersey) for the development of signal transmission over very long distances through optical fiber fibers using optical solitons. During normal transmission over fiber-optic communication lines, the signal must be amplified every 80 - 100 kilometers (the fiber itself can serve as an amplifier when it is pumped with light of a certain wavelength). And every 500 - 600 kilometers it is necessary to install a repeater that converts the optical signal into an electrical one, preserving all its parameters, and then again into an optical one for further transmission. Without these measures, the signal at a distance exceeding 500 kilometers is distorted beyond recognition. The cost of this equipment is very high: the transfer of one terabit (10 12 bits) of information from San Francisco to New York costs 200 million dollars per relay station.
The use of optical solitons, which retain their shape during propagation, makes it possible to carry out completely optical signal transmission over distances of up to 5–6 thousand kilometers. However, there are significant difficulties in the way of creating a "soliton line", which have been overcome only very recently.
The possibility of the existence of solitons in an optical fiber was predicted in 1972 by the theoretical physicist Akira Hasegawa, an employee of the Bell company. But at that time, there were no optical fibers with low losses in those wavelength regions where solitons could be observed.
Optical solitons can propagate only in a light guide with a small but finite dispersion value. However, an optical fiber that maintains the required dispersion value over the entire spectral width of a multichannel transmitter simply does not exist. And this makes "ordinary" solitons unsuitable for use in networks with long transmission lines.
A suitable soliton technology has been created over a number of years under the direction of Lynn Mollenauer, a leading specialist in the Optical Technology Department of the same Bell company. This technology was based on the development of dispersion-controlled optical fibers, which made it possible to create solitons whose pulse shape can be maintained indefinitely.
The control method is as follows. The amount of dispersion along the length of the optical fiber periodically changes between negative and positive values. In the first section of the light guide, the pulse expands and shifts in one direction. In the second section, which has a dispersion of the opposite sign, the pulse is compressed and shifted in the opposite direction, as a result of which its shape is restored. With further movement, the impulse expands again, then enters the next zone, which compensates for the action of the previous zone, and so on - a cyclic process of expansions and contractions occurs. The pulse experiences a pulsation in width with a period equal to the distance between the optical amplifiers of a conventional light guide - from 80 to 100 kilometers. As a result, according to Mollenauer, a signal with an information volume of more than 1 terabit can travel at least 5-6 thousand kilometers without retransmission at a transmission rate of 10 gigabits per second per channel without any distortion. Such a technology for ultra-long distance communication over optical lines is already close to the implementation stage.
Doctor technical sciences A. Golubev
"Science and Life" No. 11, 2001, pp. 24 - 28
http://razumru.ru
SOLITON this is a solitary wave in media of various physical nature, which retains its shape and speed unchanged during propagation. From English. solitary solitary (solitary wave solitary wave), “-on” a typical ending of terms of this kind (for example, electron, photon, etc.), meaning the likeness of a particle.
The concept of a soliton was introduced in 1965 by the Americans Norman Zabuski and Martin Kruskal, but the British engineer John Scott Russell (18081882) is credited with the discovery of the soliton. In 1834, he first described the observation of a soliton ("large solitary wave"). At that time, Russell was studying the capacity of the Union Canal near Edinburgh (Scotland). Here is how the author of the discovery himself spoke about him: “I was following the movement of a barge, which was quickly pulled along a narrow channel by a pair of horses, when the barge suddenly stopped; but the mass of water which the barge set in motion did not stop; instead, it gathered near the prow of the ship in a state of frenzied motion, then suddenly left it behind, rolling forward at great speed and taking the form of a large single elevation, i.e. rounded, smooth and well-defined water hill, which continued its path along the canal, not changing its shape in the least and without slowing down. I followed him on horseback, and when I overtook him he was still rolling forward at about eight or nine miles an hour, retaining his original elevation profile, about thirty feet long and a foot to a foot and a half high. Its height gradually decreased, and after a mile or two of pursuit I lost it in the bends of the canal. Thus, in August 1834, for the first time, I had the opportunity to encounter an extraordinary and beautiful phenomenon, which I called a wave of translation ... ".
Subsequently, Russell experimentally, after conducting a series of experiments, found the dependence of the speed of a solitary wave on its height (the maximum height above the level of the free water surface in the channel).
Perhaps Russell foresaw the role played by solitons in modern science. In the last years of his life, he completed a book Waves of translation in water, air and ethereal oceans published posthumously in 1882. This book contains a reprint Wave Reports the first description of a solitary wave, and a number of guesses about the structure of matter. In particular, Russell believed that sound is solitary waves (in fact, this is not so), otherwise, in his opinion, sound propagation would occur with distortions. Based on this hypothesis and using the dependence of the speed of a solitary wave found by him, Russell found the thickness of the atmosphere (5 miles). Moreover, making the assumption that light is also solitary waves (which is also not true), Russell also found the length of the universe (5 10 17 miles).
Apparently, in his calculations regarding the size of the universe, Russell made a mistake. However, the results obtained for the atmosphere would be correct if its density were uniform. Russell Wave Report is now considered an example of clarity in the presentation of scientific results, a clarity to which many scientists today are far away.
The reaction to Russell's scientific message of the then most respected English mechanics George Bidel Airy (18011892) (professor of astronomy at Cambridge from 1828 to 1835, astronomer of the royal court from 1835 to 1881) and George Gabriel Stokes (18191903) (professor of mathematics at Cambridge from 1849 to 1903) was negative. Many years later, the soliton was rediscovered under very different circumstances. Interestingly, it was not easy to reproduce Russell's observation. The participants of the Soliton-82 conference, who came to Edinburgh for a conference dedicated to the centenary of Russell's death and tried to get a solitary wave at the very place where Russell observed it, failed to see anything, with all their experience and extensive knowledge about solitons .
In 18711872, the results of the French scientist Joseph Valentin Boussinesq (18421929) were published, devoted to theoretical studies of solitary waves in channels (similar to the Russell solitary wave). Boussinesq got the equation:
Describing such waves ( u displacement of the free water surface in the channel, d channel depth, c 0 wave speed, t time, x spatial variable, the index corresponds to differentiation with respect to the corresponding variable), and determined their form (hyperbolic secant, cm. rice. 1) and speed.
Boussinesq called the investigated waves buckling and considered buckling of positive and negative heights. Boussinesq substantiated the stability of positive swellings by the fact that their small perturbations, having arisen, rapidly decay. In the case of negative buckling, the formation of a stable waveform is impossible, as well as for long and positive very short buckling. Somewhat later, in 1876, the Englishman Lord Rayleigh published the results of his research.
The next important stage in the development of the theory of solitons was the work (1895) of the Dutch Diederik Johann Korteweg (18481941) and his student Gustav de Vries (the exact dates of life are not known). Apparently, neither Korteweg nor de Vries have read Boussinesq's works. They derived an equation for waves in sufficiently wide channels of constant cross section, which now bears their name, the Korteweg-de Vries (KdV) equation. The solution of such an equation describes the wave discovered by Russell at the time. The main achievements of this study were to consider a simpler equation describing waves traveling in one direction, such solutions are more illustrative. Because the solution includes the Jacobi elliptic function cn, these solutions were called "cnoidal" waves.
In normal form, the KdV equation for the desired function And looks like:
The ability of a soliton to keep its shape unchanged during propagation is explained by the fact that its behavior is determined by two mutually opposite processes. Firstly, this is the so-called nonlinear steeping (the wave front of a sufficiently large amplitude tends to overturn in the areas of amplitude increase, since the rear particles, which have a large amplitude, move faster ahead of the traveling ones). Secondly, such a process as dispersion manifests itself (the dependence of the wave speed on its frequency, determined by the physical and geometric properties environment; during dispersion, different sections of the wave move at different velocities and the wave spreads). Thus, the nonlinear steepening of the wave is compensated by its spreading due to dispersion, which ensures the preservation of the shape of such a wave during its propagation.
The absence of secondary waves during the propagation of a soliton indicates that the wave energy is not scattered over space, but is concentrated in a limited space (localized). The localization of energy is a distinctive quality of the particle.
Another amazing feature of solitons (noted by Russell) is their ability to maintain their speed and shape while passing through each other. The only reminder of the interaction that has taken place are the constant displacements of the observed solitons from the positions they would have occupied if they had not met. There is an opinion that solitons do not pass through each other, but are reflected like colliding elastic balls. This also shows the analogy of solitons with particles.
For a long time it was believed that solitary waves are associated only with waves on the water and they were studied by specialists in hydrodynamics. In 1946 M.A. Lavrentiev (USSR), and in 1954 K.O. Friedrichs and D.G. Hyers of the USA published theoretical proofs of the existence of solitary waves.
The modern development of the theory of solitons began in 1955, when the work of scientists from Los Alamos (USA) Enrico Fermi, John Pasta and Stan Ulam was published, devoted to the study of nonlinear discretely loaded strings (such a model was used to study the thermal conductivity of solids). Long waves traveling along such strings turned out to be solitons. Interestingly, the research method in this work was a numerical experiment (calculations on one of the first computers created by that time).
Initially discovered theoretically for the Boussinesq and KdV equations describing waves in shallow water, solitons have now also been found as solutions to a number of equations in other areas of mechanics and physics. The most common are (below in all equations u sought functions, coefficients at u some constants)
non-linear Schrödinger equation (NLS)
The equation was obtained in the study of optical self-focusing and splitting of optical beams. The same equation was used in the study of waves in deep water. A generalization of the NSE for wave processes in plasma has appeared. It is interesting to use NSE in the theory of elementary particles.
Sin-Gordon Equation (SG)
describing, for example, the propagation of resonant ultrashort optical pulses, dislocations in crystals, processes in liquid helium, charge density waves in conductors.
Soliton solutions also have so-called related KdV equations. These equations include
modified KdV equation
equation of Benjamin, Bohn and Magoni (BBM)
first appeared in the description of the bora (waves on the surface of the water that occurs when the gates of the locks are opened, when the river is "locked");
Benjamin's equation It
obtained for waves inside a thin layer of an inhomogeneous (stratified) fluid located inside another homogeneous fluid. The study of the transonic boundary layer also leads to the Benjamin It equation.
Equations with soliton solutions also include the Born Infeld equation
having applications in field theory. There are also other equations with soliton solutions.
A soliton described by the KdV equation is uniquely characterized by two parameters: the velocity and the position of the maximum at a fixed point in time.
A soliton described by the Hirota equation
uniquely characterized by four parameters.
Since 1960, the development of the theory of solitons has been influenced by a number of physical problems. A theory of self-induced transparency was proposed and experimental results were presented to confirm it.
In 1967, Kruskal et al found a method for obtaining an exact solution of the KdV equation, the method of the so-called inverse scattering problem. The essence of the method of the inverse scattering problem is to replace the equation being solved (for example, the KdV equation) by a system of other, linear equations, the solution of which is easily found.
In 1971, the Soviet scientists V.E. Zakharov and A.B. Shabat solved the NLS by the same method.
Applications of the soliton theory are currently used in the study of signal transmission lines with nonlinear elements (diodes, resistance coils), the boundary layer, the atmospheres of planets (the Great Red Spot of Jupiter), tsunami waves, wave processes in plasma, in field theory, solid state physics , thermal physics of extreme states of substances, in the study of new materials (for example, Josephson junctions, consisting of two layers of superconducting metal separated by a dielectric), in creating models of crystal lattices, in optics, biology, and many others. It has been suggested that impulses traveling along nerves are solitons.
Currently, varieties of solitons and some combinations of them are described, for example:
antisoliton negative amplitude soliton;
breather (doublet) pair soliton antisoliton (Fig. 2);
multisoliton several solitons moving as a whole;
fluxon quantum magnetic flux, an analogue of a soliton in distributed Josephson junctions;
kink (monopole), from English kink inflection.
Formally, a kink can be introduced as a solution of the KdV, NLSE, and SG equations described by a hyperbolic tangent (Fig. 3). Reversing the sign of a kink solution gives an antikink.
Kinks were discovered in 1962 by the Englishmen Perring and Skyrme while solving the SG equation numerically (on a computer). Thus, kinks were discovered before the name soliton appeared. It turned out that the collision of kinks did not lead either to their mutual annihilation or to the subsequent appearance of other waves: kinks, thus, exhibited the properties of solitons, but the name kink was assigned to waves of this kind.
Solitons can also be two-dimensional and three-dimensional. The study of non-one-dimensional solitons was complicated by the difficulties of proving their stability, but recently experimental observations of non-one-dimensional solitons have been obtained (for example, horseshoe-shaped solitons on a film of a flowing viscous liquid, studied by V.I. Petviashvili and O.Yu. Tsvelodub). Two-dimensional soliton solutions have the Kadomtsev Petviashvili equation, which is used, for example, to describe acoustic (sound) waves:
Among the known solutions of this equation are non-spreading vortices or vortex solitons (a vortex medium is a flow of a medium in which its particles have an angular velocity of rotation about some axis). Solitons of this kind, found theoretically and modeled in the laboratory, can spontaneously arise in the atmospheres of planets. In terms of its properties and conditions of existence, a soliton-vortex is similar to a remarkable feature of Jupiter's atmosphere, the Great Red Spot.
Solitons are essentially non-linear formations and are just as fundamental as linear (weak) waves (eg sound). The creation of a linear theory, to a large extent, by the works of the classics Bernhard Riemann (18261866), Augustin Cauchy (17891857), Jean Joseph Fourier (17681830) made it possible to solve important problems facing the natural sciences of that time. With the help of solitons, it is possible to elucidate new fundamental questions when considering modern scientific problems.
Andrey Bogdanov
annotation. The report is devoted to the possibilities of the soliton approach in supramolecular biology, primarily for modeling a wide class of natural wave-like and oscillatory movements in living organisms. The author has identified many examples of the existence of soliton-like supramolecular processes ("biosolitons") in locomotor, metabolic and other phenomena of dynamic biomorphology at various lines and levels of biological evolution. Biosolitons are understood, first of all, as characteristic one-humped (unipolar) local deformations moving along the biobody with the preservation of their shape and speed.
Solitons, sometimes called "wave atoms", are endowed with properties that are unusual from the classical (linear) point of view. They are capable of acts of self-organization and self-development: self-localization; energy capture; reproduction and death; the formation of ensembles with pulsating and other dynamics. Solitons were known in plasma, liquid and solid crystals, classical liquids, nonlinear lattices, magnetic and other polydomain media, etc. The discovery of biosolitons indicates that, due to its mechanochemistry, living matter is a soliton medium with various physiological uses of soliton mechanisms. A research hunt in biology is possible for new types of solitons - breathers, wobblers, pulsons, etc., deduced by mathematicians at the "tip of a pen" and only then discovered by physicists in nature. The report is based on the monographs: S.V. Petukhov “Biosolitons. Fundamentals of soliton biology”, 1999; S.V. Petukhov "Biperiodic table genetic code and the number of protons”, 2001.
Solitons are an important object of modern physics. Intensive development Their theory and applications began after the publication in 1955 by Fermi, Pasta and Ulam of a work on computer calculation of oscillations in a simple nonlinear system of a chain of weights connected by nonlinear springs. Soon, the necessary mathematical methods were developed to solve soliton equations, which are non-linear partial differential equations. Solitons, sometimes called "wave atoms", have the properties of waves and particles at the same time, but are not in the full sense of the one or the other, but constitute a new object of mathematical natural science. They are endowed with properties that are unusual from the classical (linear) point of view. Solitons are capable of acts of self-organization and self-development: self-localization; capturing the energy coming from outside into the "soliton" medium; reproduction and death; the formation of ensembles with non-trivial morphology and dynamics of a pulsating and other character; self-complication of these ensembles when additional energy enters the medium; overcoming the tendency to disorder in soliton media containing them; etc. They can be interpreted as a specific form of organization of physical energy in matter, and accordingly, one can speak of "soliton energy" by analogy with the well-known expressions "wave energy" or "vibrational energy". Solitons are realized as states of special nonlinear media (systems) and have fundamental differences from ordinary waves. In particular, solitons are often stable self-trapped bunches of energy with a characteristic shape of a single-humped wave moving with the same shape and speed without dissipating its energy. Solitons are capable of non-destructive collisions, i.e. able to pass through each other when meeting without breaking their shape. They have numerous applications in engineering.
A solitary is usually understood as a solitary wave-like object (a localized solution of a nonlinear partial differential equation belonging to a certain class of so-called soliton equations), which is able to exist without dissipation of its energy and, when interacting with other local perturbations, always restores its original form, i.e. . capable of non-destructive collisions. As is known, soliton equations “arise in the most natural way in the study of weakly nonlinear dispersion systems of various types on various spatial and temporal scales. The universality of these equations turns out to be so striking that many were inclined to see something magical in it ... But this is not so: dispersive weakly damped or undamped nonlinear systems behave the same, regardless of whether they are encountered in the description of plasma, classical liquids, lasers or nonlinear gratings". Accordingly, solitons are known in plasma, liquid and solid crystals, classical liquids, nonlinear lattices, magnetic and other polydomain media, etc. small dissipative terms into soliton equations).
It should be noted that living matter is permeated with many non-linear lattices: from molecular polymer networks to supramolecular cytoskeletons and organic matrix. Rearrangements of these lattices are of great biological importance and may well behave in a soliton-like manner. In addition, solitons are known as forms of motion of phase rearrangement fronts, for example, in liquid crystals (see, for example, ). Since many systems of living organisms (including liquid crystal systems) exist on the verge of phase transitions, it is natural to assume that the fronts of their phase rearrangements in organisms will also often move in a soliton form.
Even the discoverer of solitons, Scott Russell, experimentally showed in the last century that the soliton acts as a concentrator, trap and transporter of energy and matter, capable of non-destructive collisions with other solitons and local perturbations. Obviously, these features of solitons can be beneficial for living organisms, and therefore biosoliton mechanisms can be specially cultivated in wildlife by the mechanisms of natural selection. Here are some of these benefits:
- - 1) spontaneous capture of energy, matter, etc., as well as their spontaneous local concentration (self-localization) and careful, loss-free transportation in a dosed form inside the body;
- - 2) ease of control over the flows of energy, matter, etc. (when they are organized in a soliton form) due to the possible local switching of the characteristics of the nonlinearity of the biomedium from the soliton to the non-soliton type of nonlinearity and vice versa;
- - 3) decoupling for many of those simultaneously and in one place occurring in the body, i.e. overlapping processes (locomotor, blood supply, metabolic, growth, morphogenetic, etc.) that require relative independence of their course. This decoupling can be ensured precisely by the ability of solitons to nondestructive collisions.
For the first time, our study of supramolecular cooperative processes in living organisms from the soliton point of view revealed the presence in them of many macroscopic soliton-like processes. The subject of study was, first of all, directly observed locomotor and other biological movements, the high energy efficiency of which had long been assumed by biologists. At the first stage of the study, we found that in many living organisms, biological macromotions often have a soliton-like appearance of a characteristic single-humped local deformation wave moving along a living body with the preservation of its shape and speed and sometimes demonstrating the ability to non-destructive collisions. These "biosolitons" are realized at various branches and levels of biological evolution in organisms that differ in size by several orders of magnitude.
The report presents numerous examples of such biosolitons. In particular, an example of crawling of the Helix snail, which occurs due to the passage of a single-humped undulating deformation along its body with the preservation of its shape and speed, is considered. Detailed registrations of this kind of biological movement are taken from the book. In one variant of crawling (with one “gait”), the snail realizes local stretching deformations that run along the supporting surface of its body from front to back. In another, slower variant of crawling, local compression deformations occur along the same bodily surface, going in the opposite direction from the tail to the head. Both named types of soliton deformations direct and retrograde can occur in the cochlea simultaneously with head-on collisions between them. We emphasize that their collision is non-destructive, which is characteristic of solitons. In other words, after a collision, they retain their shape and speed, that is, their individuality: “the presence of large retrograde waves does not affect the propagation of normal and much shorter direct waves; both types of waves propagated without any sign of mutual interference. This biological fact has been known since the beginning of the century, although researchers have never associated it with solitons before us.
As Gray and other classics of the study of locomotion (spatial movements in organisms) emphasized, the latter are highly energy-efficient processes. This is essential for the vitally important provision of the organism with the ability to move without fatigue over long distances in search of food, escape from danger, etc. (organisms in general are extremely careful with energy, which is not at all easy for them to store). Thus, in a snail, a soliton local deformation of the body, due to which its body moves in space, occurs only in the zone of separation of the body from the support surface. And the entire part of the body in contact with the support is undeformed and rests relative to the support. Accordingly, during the entire time of soliton-like deformation flowing through the body of the cochlea, such a wave-like locomotion (or mass transfer process) does not require energy costs to overcome the forces of friction of the cochlea against the support, being the most economical in this respect. Of course, it can be assumed that part of the energy during locomotion is still dissipated into the mutual friction of tissues inside the body of the cochlea. But if this locomotor wave is soliton-like, then it also ensures the minimization of friction losses inside the body. (As far as we know, the issue of energy losses due to intra-body friction during locomotion has not been sufficiently studied experimentally, however, the body is unlikely to have missed the opportunity to minimize them). With the considered organization of locomotion, all (or almost all) energy costs for it are reduced to the costs of the initial creation of each such soliton-like local deformation. It is the physics of solitons that provides extremely energy-efficient possibilities for handling energy. And its use by living organisms looks natural, especially since the surrounding world is saturated with soliton media and solitons.
It should be noted that, at least since the beginning of the century, researchers have represented wave-like locomotion as a kind of relay process. At that time of “presoliton physics”, the natural physical analogy of such a relay process was the combustion process, in which local bodily deformation was transferred from point to point like ignition. This idea of relay-race dissipative processes of the type of combustion, now called autowave, was the best possible at that time, and it has long become familiar to many. However, physics itself did not stand still. In recent decades, it has developed the idea of solitons as a new type of non-dissipative relay processes of higher energy efficiency with paradoxical properties that were previously unthinkable, which provides the basis for a new class of nonlinear models of relay processes.
One of the important advantages of the soliton approach over the traditional autowave approach in modeling processes in a living organism is determined by the ability of solitons to nondestructive collisions. Indeed, autowaves (describing, for example, the movement of the combustion zone along a burning cord) are characterized by the fact that behind them there remains a zone of non-excitability (a burnt cord), and therefore two autowaves, when colliding with each other, cease to exist, being unable to move along the already “burned out” site." But in the areas of a living organism, many biomechanical processes simultaneously occur - locomotor, blood supply, metabolic, growth, morphogenetic, etc., and therefore, modeling them with autowaves, the theorist is faced with the following problem of mutual destruction of autowaves. One autowave process, moving through the area of the body under consideration due to the continuous burning of energy reserves on it, makes this medium unexcitable for other autowaves for some time until the energy reserves for their existence are restored in this area. In living matter, this problem is especially relevant also because the types of energy-chemical reserves in it are highly unified (organisms have a universal energy currency - ATP). Therefore, it is difficult to believe that the fact of the simultaneous existence of many processes in one area in the body is ensured by the fact that each autowave process in the body moves by burning out its specific type of energy without burning out energy for others. For soliton models, this problem of mutual annihilation of biomechanical processes colliding in one place does not exist in principle, since solitons, due to their ability to non-destructive collisions, calmly pass through each other and their number can be arbitrarily large in one area at the same time. According to our data, the soliton sine-Gordon equation and its generalizations are of particular importance for modeling biosoliton phenomena of living matter.
As is known, in polydomain media (magnets, ferroelectrics, superconductors, etc.), solitons act as interdomain walls. In living matter, the polydomain phenomenon plays an important role in morphogenetic processes. As in other polydomain environments, in polydomain biological environments it is associated with classical principle Landau-Lifshitz minimization of energy in a medium. In these cases, soliton interdomain walls turn out to be places of increased energy concentration, in which biochemical reactions often proceed especially actively.
The ability of solitons to play the role of locomotives transporting portions of matter in Right place within the soliton environment (organism) according to the laws of nonlinear dynamics, also deserves every attention in connection with bioevolutionary and physiological problems. Let us add that the biosoliton physical energy is able to harmoniously coexist in a living organism with the known chemical types of its energy. The development of the concept of biosolitons makes it possible, in particular, to open a research “hunt” in biology for analogues different types solitons breathers, wobblers, pulsons, etc., deduced by mathematicians "on the tip of a pen" when analyzing soliton equations and then discovered by physicists in nature. Many oscillatory and wave physiological processes can eventually receive meaningful soliton models for their description, associated with the nonlinear, soliton nature of a biopolymer living substance.
For example, this refers to the basic physiological movements of a living biopolymer substance such as heartbeats, etc. Recall that in a human embryo at the age of three weeks, when it has a growth of only four millimeters, the heart comes first in motion. The beginning of cardiac activity is due to some internal energy mechanisms, since at this time the heart does not yet have any nerve connections to control these contractions and it begins to contract when there is still no blood to be pumped. At this moment, the embryo itself is essentially a piece of polymeric mucus, in which internal energy is self-organizing into energy-efficient pulsations. The same can be said about the occurrence of heartbeats in the eggs and eggs of animals, where the supply of energy from the outside is minimized by the existence of the shell and other insulating covers. Similar forms of energy self-organization and self-localization are known in polymeric media, including those of a non-biological type, and, according to modern concepts, are of a soliton nature, since solitons are the most energy-efficient (non-dissipative or low-dissipative) self-organizing structures of a pulsating and other nature. Solitons are realized in a variety of natural media surrounding living organisms: solid and liquid crystals, classical liquids, magnets, lattice structures, plasma, etc. The evolution of living matter with its natural selection mechanisms has not passed by the unique properties of solitons and their ensembles.
Do these materials have anything to do with synergy? Yes, definitely. As defined in Hagen's monograph /6, p.4/, “within the framework of synergetics, such a joint action of individual parts of any disordered system is studied, as a result of which self-organization occurs - macroscopic spatial, temporal or space-time structures arise, and are considered as deterministic and stochastic processes. There are many types of nonlinear processes and systems that are studied within the framework of synergetics. Kurdyumov and Knyazeva /7, p.15/, listing a number of these types, specifically note that among them one of the most important and intensively studied are solitons. In recent years, the international journal Chaos, Solitons & Fractals has been published. Solitons observed in various natural environments are a vivid example of the nonlinear cooperative behavior of many elements of the system, leading to the formation of specific spatial, temporal, and spatiotemporal structures. The most well-known, although by no means the only, type of such soliton structures is the self-localizing one-humped local deformation of the medium, which is stable in shape and runs at a constant speed. Solitons are actively used and studied in modern physics. Since 1973, starting from the works of Davydov /8/, solitons are also used in biology for modeling molecular biological processes. At present, there are many publications all over the world on the use of such "molecular solitons" in molecular biology, in particular, for understanding the processes in proteins and DNA. Our works /3, 9/ were the first publications in the world literature on the topic of "supramolecular solitons" in biological phenomena of the supramolecular level. We emphasize that the existence of molecular biosolitons (which, according to many authors, has yet to be proved) does not in any way imply the existence of solitons in cooperative biological supramolecular processes that unite myriads of molecules.
LITERATURE:
- Dodd R. et al. Solitons and nonlinear wave equations. M., 1988, 694 p.
- Kamensky V.G. ZhETF, 1984, vol. 87, issue. 4(10), p. 1262-1277.
- Petukhov S.V. Biosolitons. Fundamentals of soliton biology. - M., 1999, 288 p.
- Gray J. Animal locomotion. London, 1968.
- Petukhov S.V. Biperiodic table of the genetic code and the number of protons. - M., 2001, 258 p.
- Hagen G. Synergetics. - M., Mir, 1980, 404 p.
- Knyazeva E.N., Kurdyumov S.P. Laws of evolution and self-organization of complex systems. M., Nauka, 1994, 220 p.
- Davydov A.S. Solitons in biology. - Kyiv, Naukova Dumka, 1979.
- Petukhov S.V. Solitons in biomechanics. Deposited at VINITI RAS on February 12, 1999, No. 471-B99. (Index VINITI "Deposited scientific works", No. 4 for 1999)
Summary
. The report discusses the opportunities opened up by a solitonic approach to supramolecular biology, first of all, for modeling a wide class of natural wave movements in living organisms. The results of the author’s research demonstrate the existence of soliton-like supramolecular processes in locomotor, metabolic and other manifestations of dynamic biomorphology on a wide variety of branches and levels of biological evolution.
Solitons, named sometimes "wave atoms", have unusual properties from the classical (linear) viewpoint. They have the ability for self-organizing: auto-localizations; catching of energy; formation of ensembles with dynamics of pulsing and other character. Solitons were known in plasma, liquid and firm crystals, classical liquids, nonlinear lattices, magnetic and other poly-domain matters, etc. The reveal of biosolitons points out that biological mechano-chemistry makes living matter as a solitonic environment with opportunities of various physiological use of solitonic mechanisms. The report is based on the books: S.V. Petoukhov "Biosolitons. Bases of solitonic biology”, Moscow, 1999 (in Russian).
Petukhov S.V., Solitons in cooperative biological processes of the supramolecular level // "Academy of Trinitarianism", M., El No. 77-6567, publ. 13240, 21.04.2006
In the new edition (1st ed. - 1985), the material of the book has been substantially revised taking into account the latest achievements.
For high school students, students, teachers.
Preface to the first edition 5
Preface to the second edition 6
Introduction 7
Part I. HISTORY OF THE SOLITON 16
Chapter 1. 150 years ago 17
The beginning of the theory of waves (22). The Weber brothers study waves (24). On the usefulness of the theory of waves (25). About the main events of the era (28). Science and Society (34).
Chapter 2
Until the fatal meeting (38). Encounter with a solitary wave (40). It can't be! (42). And yet it exists! (44). Solitary Wave Rehabilitation (46). Solitary Wave Isolation (49). Wave or particle? (50).
Chapter 3. Relatives of the soliton 54
Hermann Helmholtz and the nerve impulse (55). Further fate of the nerve impulse (58). Hermann Helmholtz and whirlwinds (60). "Vortex atoms" Kelvin (68). Lord Ross and whirlwinds in space (69). On linearity and non-linearity (71).
Part II. NONLINEAR OSCILLATIONS AND WAVES 76 Chapter 4. Portrait of a pendulum 77
Pendulum equation (77). Small oscillations of the pendulum (79). Galileo's pendulum (80). On similarity and dimensions (82). Conservation of energy (86). Language of phase diagrams (90). Phase portrait (97). Phase portrait of the pendulum (99). "Soliton" solution of the pendulum equation (103). Pendulum motions and "manual" soliton (104). Concluding remarks (107).
Waves in a chain of coupled particles (114). Retreat into history. The Bernoulli family and the waves (123). Waves of D'Alembert and disputes around them (125). On discrete and continuous (129). How the speed of sound was measured (132). Dispersion of waves in a chain of atoms (136). How to "hear" the Fourier expansion? (138). A few words about the dispersion of light (140). Dispersion of waves on water (142). How fast does a flock of waves run (146). How much energy is in the wave (150).
Part III. THE PRESENT AND FUTURE OF SOL EETON 155
What is theoretical physics (155). Ideas of Ya. I. Frenkel (158). Atomic model of a moving dislocation according to Frenkel and Kontorova (160). Interaction of dislocations (164). "Living" soliton atom (167). Dialogue between the reader and the author (168). Dislocations and pendulums (173). What sound waves have turned into (178). How to see dislocations? (182). Desktop solitons (185). Other close relatives of dislocations along the mathematical line (186). Magnetic solitons (191).
Can a person "be friends" with a computer (198). Many-faced chaos (202). Computer surprises Enrico Fermi (209) Return of Russell's soliton (215). Oceanic solitons: tsunami, "ninth wave" (227). Three solitons (232). Soliton telegraph (236). The nerve impulse is the "elementary particle" of thought (241). Omnipresent whirlwinds (246). Josephson effect (255). Solitons in long Josephson junctions (260). Elementary particles and solitons (263). Unified theories and strings (267).
Chapter 6 Frenkel Solitons 155
Chapter 7. Rebirth of the soliton 195
Applications
Short name index
Many people have probably come across the word "co-lithon", consonant with such words as electron or proton. This book is dedicated to the scientific idea behind this easily remembered word, its history and creators.
It is designed for the widest range of readers who have mastered the school course in physics and mathematics and who are interested in science, its history and applications. Not everything is told in it about solitons. But most of what remained after all the restrictions, I tried to set out in sufficient detail. At the same time, some well-known things (for example, about oscillations and waves) had to be presented somewhat differently than was done in other popular science and quite scientific books and articles, which I, of course, widely used. It is quite impossible to list their authors and mention all the scientists whose conversations influenced the content of this book, and I offer them my apologies along with deep gratitude.
I would especially like to thank S. P. Novikov for constructive criticism and support, L. G. Aslamazov and Ya. A. Smorodinsky for valuable advice, as well as Yu. S. Galpern and S. R. Filonovich, who carefully read the manuscript and made many comments that contributed to its improvement.
This book was written in 1984, and when preparing a new edition, the author, naturally, wanted to talk about new interesting ideas that have been born recently. The main additions relate to optical and Josephson solitons, the observation and application of which have recently been the subject of very interesting papers. The section devoted to chaos has been somewhat expanded, and on the advice of the late Yakov Borisovich Zel'dovich, shock waves and detonation are described in more detail. An essay on modern unified theories of particles and their interactions is added at the end of the book. It also attempts to give some idea of relativistic strings - a new and rather mysterious physical object, with the study of which hopes are associated for creating a unified theory of all interactions known to us. A small math appendix has been added, as well as a short name index.
The book also has a lot of smaller changes - something thrown out, and something added. It hardly needs to be described in detail. The author tried to greatly expand everything related to computers, but this idea had to be abandoned, it would be better to devote a separate book to this topic. I hope that the enterprising reader, armed with some kind of computer, will be able to invent and implement his own computer experiments on the material of this book.
In conclusion, I am pleased to express my gratitude to all readers of the first edition, who provided their comments and suggestions on the content and form of the book. I have tried to accommodate them to the best of my ability.
Nowhere is the unity of nature and the universality of its laws manifested so clearly as in oscillatory and wave phenomena. Each student can easily answer the question: “What is common between a swing, a clock, a heart, an electric bell, a chandelier, a TV, a saxophone and an ocean liner?” - and easily continue this list. The common thing, of course, is that oscillations exist or can be excited in all these systems.
Some of them we see with the naked eye, others are observed with the help of instruments. Some oscillations are very simple, such as swing oscillations, others are much more complicated - just look at electrocardiograms or encephalograms, but we can always easily distinguish an oscillatory process by its characteristic repetition, periodicity.
We know that a wobble is a periodic movement or change of state, no matter what moves or changes state. The science of fluctuations studies what is common in vibrations of very different nature.
In the same way, waves of a completely different nature can be compared - ripples on the surface of a puddle, radio waves, a "green wave" of traffic lights on a highway - and many, many others. The science of waves studies the waves themselves, abstracting from their physical nature. A wave is considered as a process of excitation transfer (in particular, oscillatory motion) from one point of the medium to another. In this case, the nature of the medium and the specific nature of its excitations are unimportant. Therefore, it is natural that oscillatory and sound waves and the connections between them are studied today by a single science - theory
vibrations and waves. The general nature of these links is well known. The clock ticks, the bell rings, the swing swings and creaks, emitting sound waves; a wave propagates through the blood vessels, which we observe by measuring the pulse; electromagnetic oscillations excited in the oscillatory circuit are amplified and carried away into space in the form of radio waves; "oscillations" of electrons in atoms give rise to light, etc.
When a simple periodic wave of small amplitude propagates, the particles of the medium perform periodic motions. With a small increase in the amplitude of the wave, the amplitude of these movements also increases proportionally. If, however, the amplitude of the wave becomes large enough, new phenomena may arise. For example, waves on water at high altitude become steep, breakers form on them, and they eventually capsize. In this case, the nature of the motion of the particles of the wave completely changes. Water particles in the crest of a wave begin to move completely randomly, i.e., regular, oscillatory motion turns into irregular, chaotic. This is the most extreme degree of manifestation of the nonlinearity of waves on the water. A weaker manifestation of nonlinearity is the dependence of the waveform on its amplitude.
To explain what non-linearity is, one must first explain what linearity is. If the waves have a very small height (amplitude), then with an increase in their amplitude, say, by a factor of two, they remain exactly the same, their shape and propagation speed do not change. If one such wave runs into another, then the resulting more complex movement can be described by simply adding the heights of both waves at each point. A well-known explanation of the phenomenon of wave interference is based on this simple property of linear waves.
Waves with sufficiently small amplitude are always linear. However, as the amplitude increases, their shape and speed begin to depend on the amplitude, and they can no longer simply be added, the waves become nonlinear. At a large amplitude, the nonlinearity generates breakers and leads to wave breaking.
The shape of the waves can be distorted not only because of the non-linearity. It is well known that waves of different lengths propagate, generally speaking, with different speeds. This phenomenon is called dispersion. Observing waves running in circles from a stone thrown into the water, it is easy to see that long waves on the water run faster than short ones. If a slight elevation has formed on the surface of the water in a long and narrow groove (it is easy to make it with the help of partitions that can be quickly removed), then, thanks to dispersion, it will quickly break up into separate waves of different lengths, dissipate and disappear.
It is remarkable that some of these water mounds do not disappear, but live long enough to retain their shape. It is not at all easy to see the birth of such unusual "solitary" waves, but nevertheless, 150 years ago, they were discovered and studied in experiments, the idea of which has just been described. The nature of this amazing phenomenon has long remained mysterious. It seemed that it contradicted the well-established laws of the formation and propagation of waves by science. Only many decades after the publication of the report on experiments with solitary waves, their riddle was partially solved. It turned out that they can form when the effects of non-linearity, which make the mound steeper and tend to overturn it, and the effects of dispersion, which make it flatter and tend to blur it, "balance" out. Between the Scylla of nonlinearity and the Charybdis of dispersion, solitary waves are born, most recently called solitons.
Already in our time, the most amazing properties of solitons were discovered, thanks to which they became the subject of fascinating scientific research. They will be discussed in detail in this book. One of the remarkable properties of a solitary wave is that it is like a particle. Two solitary waves can collide and move apart like billiard balls, and in some cases one can think of a soliton as simply a particle whose motion obeys Newton's laws. The most remarkable thing about the soliton is its diversity. Over the past 50 years, many solitary waves have been discovered and studied, similar to solitons on the wave surface, but existing under completely different conditions.
Their common nature became clear relatively recently, in the last 20-25 years.
Now solitons are being studied in crystals, magnetic materials, superconductors, in living organisms, in the atmosphere of the Earth and other planets, in galaxies. Apparently, solitons played an important role in the evolution of the Universe. Many physicists are now fascinated by the idea that elementary particles (such as the proton) can also be considered as solitons. Modern theories of elementary particles predict various solitons that have not yet been observed, for example, solitons that carry a magnetic charge!
The use of solitons for storing and transmitting information is already beginning. The development of these ideas in the future may lead to revolutionary changes, for example, in communication technology. In general, if you have not heard about solitons, you will hear very soon. This book is one of the first attempts to explain solitons in an accessible way. Of course, it is impossible to talk about all the solitons known today, and it is not worth even trying. Yes, this is not necessary.
Indeed, in order to understand what oscillations are, it is not at all necessary to get acquainted with the whole variety of oscillatory phenomena occurring in nature and. technique. It is enough to understand the basic ideas of the science of vibrations on the simplest examples. For example, all small oscillations are similar to each other, and it is enough for us to understand how a weight on a spring or a pendulum in a wall clock oscillates. The simplicity of small oscillations is related to their linearity - the force that returns the weight or pendulum to the equilibrium position is proportional to the deviation from this position. An important consequence of linearity is the independence of the oscillation frequency from their amplitude (range).
If the linearity condition is violated, then the oscillations are much more diverse. Nevertheless, some types of non-linear oscillations can be distinguished, having studied which, one can understand the work of the most different systems- clock, heart, saxophone, generator of electromagnetic oscillations...
The most important example of non-linear oscillations is given by the movements of the same pendulum, if we do not limit ourselves to small amplitudes and arrange the pendulum so that it can not only swing, but also rotate. It is remarkable that, having dealt well with the pendulum, one can also understand the structure of the soliton! It is on this path that we, the reader, will try to understand what a soliton is.
Although this is the easiest road to the country where solitons live, many difficulties lie in wait for us on it, and one who wants to truly understand the soliton must be patient. First you need to study the linear oscillations of the pendulum, then to understand the connection between these oscillations and linear waves, in particular to understand the nature of the dispersion of linear waves. It's not that hard. The relationship between non-linear oscillations and non-linear waves is much more complex and subtle. But still, we will try to describe it without complicated mathematics. We are able to adequately represent only one type of solitons, while the rest will have to be dealt with by analogy.
Let the reader perceive this book as a journey to unfamiliar lands, in which he will get to know one city in detail, and walk around the rest of the places, looking at everything new and trying to connect it with what he has already managed to understand. You still need to get to know one city well enough, otherwise there is a risk of missing out on the most interesting because of ignorance of the language, customs and customs of foreign lands.
So, on the road, reader! Let this "collection of motley chapters" be a guide to an even more motley and diverse country where oscillations, waves and solitons live. To facilitate the use of this guide, we first need to say a few words about what it contains, what it does not contain.
Going to an unfamiliar country, it is natural to first get acquainted with its geography and history. In our case, this is almost the same, since the study of this country, in fact, is just beginning, and we do not even know its exact borders.
The first part of the book outlines the history of the solitary wave along with the basic ideas about it. Then things are told about things that at first glance are quite unlike a solitary wave on the surface of the water - about vortices and a nerve impulse. Their study also began in the last century, but the relationship with solitons was established quite recently.
The reader can really understand this connection if he has the patience to get to the last chapter. By compensating for the efforts expended, he will be able to see the deep inner relationship of such dissimilar phenomena as tsunamis, forest fires, anticyclones, sunspots, hardening of metals during forging, magnetization of iron, etc.
But first, we will have to plunge into the past for a while, to the first half of the 19th century, when ideas arose that were fully mastered only in our time. In this past, we will be primarily interested in the history of the doctrine of oscillations and waves and how, against this background, ideas arose, developed and were perceived, which later formed the foundation of the science of solitons. We will be interested in the fate of ideas, and not the fate of their creators. As Albert Einstein said, the history of physics is a drama, a drama of ideas. In this drama, “... it is instructive to follow the changing fate of scientific theories. They are more interesting than the changing destinies of people, because each of them includes something immortal, at least a particle of eternal truth.
*) These words belong to the Polish physicist Marian Smoluchowski, one of the creators of the theory brownian motion. The reader can follow the development of some basic physical ideas (such as wave, particle, field, relativity) from the wonderful popular book "Evolution of Physics" by A. Einstein and T. Infeld (Moscow: GTTI, 1956).
Nevertheless, it would be wrong not to mention the creators of these ideas, and in this book a lot of attention is paid to people who first expressed certain valuable thoughts, regardless of whether they became famous scientists or not. The author especially tried to extract from oblivion the names of people who were not sufficiently appreciated by their contemporaries and descendants, as well as to recall some little-known works of quite famous scientists. (Here, for example, the life of several scientists, little known to a wide circle of readers, and who expressed ideas related to the soliton in one way or another, is described; others are given only brief data.)
This book is not a textbook, much less a textbook on the history of science. Perhaps not all the historical information presented in it is presented absolutely accurately and objectively. The history of the theory of oscillations and waves, especially nonlinear ones, has not been sufficiently studied. The history of solitons has not yet been written at all. Perhaps the pieces of the puzzle of this story, collected by the author in different places, will be useful to someone for a more serious study. In the second part of the book, we will mainly focus on the physics and mathematics of nonlinear oscillations and waves in the form and volume in which this is necessary for a sufficiently deep acquaintance with the soliton.
The second part has a relatively large amount of mathematics. It is assumed that the reader has a fairly good understanding of what a derivative is and how speed and acceleration are expressed using the derivative. It is also necessary to remember some trigonometry formulas.
You can’t do without mathematics at all, but in fact we will need a little more than what Newton knew. Two hundred years ago, Jean Antoine Condorcet, a French philosopher, educator and one of the reformers of school teaching, said: “At present, a young man, after leaving school, knows more from mathematics than Newton acquired by deep study or discovered by his genius; he knows how to use the tools of calculation with ease, then inaccessible. We will add to what Condorcet suggested to famous schoolchildren, a few of the achievements of Euler, the Bernoulli family, d'Alembert, Lagrange and Cauchy. This is quite enough to understand modern physical concepts of a soliton. About modern mathematical theory solitons is not described - it is very complicated.
Nevertheless, in this book we will recall everything that is needed from mathematics, and, in addition, the reader who does not want or has no time to understand the formulas can simply skim through them, following only physical ideas. Things that are more difficult or take the reader away from the main road are in small print.
The second part gives some idea of the doctrine of vibrations and waves, but it does not talk about many important and interesting ideas. On the contrary, what is needed to study solitons is described in detail. The reader who wants to get acquainted with the general theory of oscillations and waves should look into other books. Solitons are associated with such different
sciences, that the author had in many cases to recommend other books for a more detailed acquaintance with some phenomena and ideas, which are mentioned here too briefly. In particular, it is worth looking into other issues of the Kvant Library, which are often cited.
The third part tells in detail and consistently about one type of solitons, which entered science 50 years ago, regardless of the solitary wave on the wom, and is associated with dislocations in crystals. The last chapter shows how the fates of all solitons eventually crossed and a general idea of solitons and soliton-like objects was born. Computers played a special role in the birth of these general ideas. Computer calculations, which led to the second birth of the soliton, were the first example of a numerical experiment, when computers were used not just for calculations, but to discover new phenomena unknown to science. Numerical experiments on computers undoubtedly have a great future, and they are described in sufficient detail.
After that, we turn to a story about some modern ideas about solitons. Here the exposition gradually becomes more and more concise, and the last paragraphs of Chap. 7 give only a general idea of the directions in which the science of solitons is developing. The purpose of this very short excursion is to give an idea of the science of today and a little look into the future.
If the reader is able to catch the internal logic and unity in the motley picture presented to him, then the main goal that the author set for himself will be achieved. The specific task of this book is to tell about the soliton and its history. The fate of this scientific idea in many respects seems unusual, but upon deeper reflection it turns out that many scientific ideas that today constitute our common wealth were born, developed and perceived with no less difficulties.
From this arose the broader task of this book - using the example of a soliton, to try to show how science works in general, how it eventually gets to the truth after many misunderstandings, misconceptions and mistakes. The main goal of science is to obtain true and complete knowledge about the world, and it can benefit people only to the extent that it approaches this goal. The most difficult thing here is completeness. Truth scientific theory we eventually establish through experimentation. However, no one can tell us how to come up with a new scientific idea, a new concept, with the help of which whole worlds of phenomena, previously separated, or even completely eluding our attention, enter the sphere of harmonious scientific knowledge. One can imagine a world without solitons, but it will already be a different, poorer world. The idea of a soliton, like other big scientific ideas, is valuable not only because it brings benefits. It further enriches our perception of the world, revealing its inner beauty that eludes a superficial glance.
The author especially wanted to reveal to the reader this side of the scientist's work, which relates it to the work of a poet or composer, who reveals to us the harmony and beauty of the world in areas that are more accessible to our senses. The work of a scientist requires not only knowledge, but also imagination, observation, courage and dedication. Perhaps this book will help someone decide to follow the disinterested knights of science, whose ideas are described in it, or at least to reflect and try to understand what made their thought tirelessly work, never satisfied with what they have achieved. The author would like to hope so, but, unfortunately, "it is not given to us to predict how our word will respond ..." What happened from the author's intention is to judge the reader.
SOLITON HISTORY
The science! you are a child of the Gray Times!
Changing everything with the attention of transparent eyes.
Why do you disturb the poet's dream...
Edgar Poe
The first officially recorded meeting of a person with a soliton occurred 150 years ago, in August 1834, near Edinburgh. This meeting was, at first glance, accidental. A person did not specially prepare for it, and special qualities were required from him so that he could see the unusual in a phenomenon that others also encountered, but did not notice anything surprising in it. John Scott Russell (1808 - 1882) was fully endowed with just such qualities. He not only left us a scientifically accurate and vivid description of his encounter with the soliton*, not without poetry, but also devoted many years of his life to the study of this phenomenon that struck his imagination.
*) He called it a wave of translation (transfer) or a great solitary wave (great solitary wave). From the word solitary, the term "soliton" was later produced.
Russell's contemporaries did not share his enthusiasm, and the solitary wave did not become popular. From 1845 to 1965 no more than two dozen scientific papers directly related to co-lithons were published. During this time, however, close relatives of the soliton were discovered and partially studied, but the universality of soliton phenomena was not understood, and Russell's discovery was hardly remembered.
In the last twenty years, a new life of the soliton has begun, which turned out to be truly many-sided and ubiquitous. Thousands of scientific papers about solitons in physics, mathematics, hydromechanics, astrophysics, meteorology, oceanography, and biology are published annually. are going scientific conferences, specially devoted to solitons, books are written about them, all more scientists are involved in an exciting hunt for solitons. In short, the solitary wave emerged from seclusion into greater life.
How and why this amazing turn in the fate of the soliton occurred, which even Russell, who was in love with the soliton, could not foresee, the reader will find out if he has the patience to read this book to the end. In the meantime, let's try to mentally travel back to 1834 in order to imagine the scientific atmosphere of that era. This will help us better understand the attitude of Russell's contemporaries to his ideas and the further fate of the soliton. Our excursion into the past will, of necessity, be very cursory, we will get acquainted mainly with those events and ideas that turned out to be directly or indirectly connected with the soliton.
Chapter 1
150 YEARS AGO
Nineteenth century, iron,
Wonstiyu cruel age ...
A. Blok
Our poor age - how many attacks on it, what a monster they consider it! And all for the railroads, for the steamships - these are his great victories, not only over the mother, but over space and time.
V. G. Belinsky
So, the first half of the last century, the time of not only Napoleonic wars, social shifts and revolutions, but also scientific discoveries, the significance of which was revealed gradually, decades later. At that time, few people knew about these discoveries, and only a few could foresee their great role in the future of mankind. We now know about the fate of these discoveries and will not be able to fully appreciate the difficulties of their perception by contemporaries. But let's still try to strain our imagination and memory and try to break through the layers of time.
1834... There is still no telephone, radio, television, cars, planes, rockets, satellites, computers, nuclear power and much more. The first one was built just five years ago. Railway, and just started building steamboats. The main type of energy used by people is the energy of heated steam.
However, ideas are already ripening that will eventually lead to the creation of technical miracles of the 20th century. All this will take almost a hundred years. Meanwhile, science is still concentrated in universities. The time for narrow specialization has not yet come, and physics has not yet emerged as a separate science. Courses in “natural philosophy” (that is, natural science) are taught at universities, the first physical institute will be created only in 1850. At that distant time, fundamental discoveries in physics can be made by very simple means, it is enough to have a brilliant imagination, observation and golden hands.
One of the most amazing discoveries of the last century was made using a wire through which an electric current was passed, and a simple compass. It cannot be said that this discovery was completely accidental. Russell's older contemporary, Hans Christian Oersted (1777 - 1851), was literally obsessed with the idea of a connection between various natural phenomena, including between heat, sound, electricity, magnetism *). In 1820, during a lecture on the search for links between magnetism and "galvanism" and electricity, Oersted noticed that when a current is passed through a wire parallel to the compass needle, the arrow deviates. This observation aroused great interest in educated society, and in science gave rise to an avalanche of discoveries, begun by André Marie Ampère (1775 - 1836).
*) The close relationship between electrical and magnetic phenomena was first noticed at the end of the 18th century. Petersburg Academician Franz Aepinus.
In the famous series of works of 1820 - 1825. Ampere laid the foundations for a unified theory of electricity and magnetism and called it electrodynamics. Then followed the great discoveries of the brilliant self-taught Michael Faraday (1791 - 1867), made by him mainly in the 30s - 40s, from the observation of electromagnetic induction in 1831 to the formation by 1852 of the concept of an electromagnetic field. Faraday also staged his experiments, which struck the imagination of his contemporaries, using the simplest means.
In 1853, Hermann Helmholtz, who will be discussed later, writes: “I managed to get acquainted with Faraday, indeed the first physicist in England and Europe ... He is simple, amiable and unpretentious, like a child; I have never met such an endearing person... He was always helpful, showed me everything that was worth seeing. But he had to look around a little, because old pieces of wood, wire and iron serve him for his great discoveries.
At this time, the electron is still unknown. Although suspicions about the existence of an elementary electric charge appeared in Faraday already in 1834 in connection with the discovery of the laws of electrolysis, its existence became a scientifically established fact only at the end of the century, and the term “electron” itself would be introduced only in 1891.
A complete mathematical theory of electromagnetism has not yet been created. Its creator, James Clark Maxwell, was only three years old in 1834, and he is growing up in the same city of Edinburgh, where the hero of our story lectures on natural philosophy. At this time, physics, which has not yet been divided into theoretical and experimental, is only beginning to be mathematized. Thus, Faraday did not use even elementary algebra in his works. Although Maxwell would say later that he adhered "not only to the ideas, but also to the mathematical methods of Faraday," this statement can only be understood in the sense that Maxwell was able to translate Faraday's ideas into the language of contemporary mathematics. In his Treatise on Electricity and Magnetism he wrote:
“Perhaps it was a happy circumstance for science that Faraday was not actually a mathematician, although he was perfectly familiar with the concepts of space, time and force. Therefore, he was not tempted to delve into interesting but purely mathematical investigations, which his discoveries would require if they were presented in mathematical form ... Thus, he was able to go his own way and coordinate his ideas with the facts obtained, using natural, not technical language... Starting to study the work of Faraday, I found that his method of understanding phenomena was also mathematical, although not represented in the form of ordinary mathematical symbols. I also found that this method can be expressed in the usual mathematical form and thus compared with the methods of professional mathematicians.
If you ask me... will this age be called the iron age or the age of steam and electricity, I will answer without hesitation that our age will be called the age of the mechanical worldview...
At the same time, the mechanics of systems of points and solids, as well as the mechanics of the motions of fluids (hydrodynamics), had already been essentially mathematized, that is, they had largely become mathematical sciences. The problems of the mechanics of systems of points were completely reduced to the theory of ordinary differential equations (Newton's equations - 1687, the more general Lagrange equations - 1788), and the problems of hydromechanics - to the theory of the so-called differential equations with partial derivatives (Euler's equations - 1755). , Navier equations - 1823). This does not mean that all tasks have been solved. On the contrary, deep and important discoveries were subsequently made in these sciences, the flow of which does not dry up even today. Mechanics and hydromechanics have simply reached that level of maturity when the basic physical principles were clearly formulated and translated into the language of mathematics.
Naturally, these deeply developed sciences served as the basis for constructing theories of new physical phenomena. To understand a phenomenon for a scientist of the last century meant to explain it in the language of the laws of mechanics. Celestial mechanics was considered an example of a consistent construction of a scientific theory. The results of its development were summed up by Pierre Simon Laplace (1749 - 1827) in the monumental five-volume Treatise on Celestial Mechanics, which was published in the first quarter of the century. This work, which collected and summarized the achievements of the giants of the XVIII century. - Bernoulli, Euler, D'Alembert, Lagrange and Laplace himself, had a profound influence on the formation of a "mechanical worldview" in the 19th century.
Note that in the same 1834 in a harmonious picture classical mechanics Newton and Lagrange, the final stroke was added - the famous Irish mathematician William Rowan Hamilton (1805 - 1865) gave the equations of mechanics the so-called canonical form (according to the dictionary of S. I. Ozhegov, “canonical” means “taken as a model, firmly established, corresponding to the canon”) and discovered the analogy between optics and mechanics. Hamilton's canonical equations were destined to play an outstanding role at the end of the century in the creation of statistical mechanics, and the optical-mechanical analogy, which established the connection between wave propagation and particle motion, was used in the 20s of our century by the creators of quantum theory. The ideas of Hamilton, who was the first to deeply analyze the concept of waves and particles and the connection between them, played a significant role in the theory of solitons.
The development of mechanics and hydromechanics, as well as the theory of deformations of elastic bodies (the theory of elasticity), was spurred on by the needs of developing technology. J.K. Maxwell also dealt a lot with the theory of elasticity, the theory of stability of motion with applications to the operation of regulators, and structural mechanics. Moreover, while developing his electromagnetic theory, he constantly resorted to illustrative models: “... I remain hopeful, when carefully studying the properties of elastic bodies and viscous liquids, to find a method that would allow us to give some mechanical image for the electrical state ... ( compare with the work: William Thomson "On the mechanical representation of electrical, magnetic and galvanic forces", 1847)".
Another famous Scottish physicist William Thomson (1824 - 1907), who later received scientific merit the title of Lord Kelvin, generally believed that all natural phenomena must be reduced to mechanical movements and explained in the language of the laws of mechanics. Thomson's views had a strong influence on Maxwell, especially in his younger years. It is surprising that Thomson, who knew and appreciated Maxwell closely, was one of the last to recognize his electromagnetic theory. This happened only after the famous experiments of Pyotr Nikolaevich Lebedev on measuring light pressure (1899): “I fought with Maxwell all my life ... Lebedev forced me to surrender ...”
The Beginning of Wave Theory
Although the basic equations describing the motion of a fluid, in the 30s of the XIX century. have already been obtained, the mathematical theory of waves on water has just begun to be created. The simplest theory waves on the surface of water was given by Newton in his Principia Mathematica, first published in 1687. One hundred years later, the famous French mathematician Joseph Louis Lagrange (1736 - 1813) called this work "the greatest work of the human mind." Unfortunately, this theory was based on the wrong assumption that water particles in a wave simply oscillate up and down. Despite the fact that Newton did not give a correct description of the waves on the water, he correctly set the problem, and his simple model gave rise to other studies. For the first time the correct approach to surface waves was found by Lagrange. He understood how it is possible to build a theory of waves on water in two simple cases - for waves with a small amplitude (“small waves”) and for waves in vessels, the depth of which is small compared to the wavelength (“shallow water”), Lagrange did not study detailed development of the theory of waves, as he was fascinated by other, more general mathematical problems.
Are there many people who, admiring the play of waves on the surface of a brook, think how to find equations by which one could calculate the shape of any wave crest?
Soon, an exact and surprisingly simple solution of the equations describing
waves on the water. This is the first, and one of the few exact, solution of the equations of hydromechanics was obtained in 1802 by a Czech scientist, professor of mathematics in
Prague Frantisek Josef Gerstner (1756 - 1832) *).
*) Sometimes F.I. Gerstner is confused with his son, F.A. Gerstner, who lived in Russia for several years. Under his leadership in 1836 - 1837. The first railway in Russia was built (from St. Petersburg to Tsarskoye Selo).
In the Gerstner wave (Fig. 1.1), which can only form in "deep water", when the wavelength is much less than the depth of the vessel, the fluid particles move in circles. The Gerstner wave is the first non-sinusoidal waveform studied. From the fact that LIQUID particles move in circles, it can be concluded that the surface of water has the shape of a cycloid. (from the Greek "kyklos" - a circle and "eidos" - a shape), that is, a curve that describes some point of a wheel rolling on a flat road. Sometimes this curve is called a trochoid (from the Greek "trochos" - wheel), and the Gerstner waves are called trochoidal *). Only for very small waves, when the height of the waves becomes much less than their length, the cycloid becomes similar to a sinusoid, and the Gerstner wave turns into a sinusoid. Although the particles of water deviate little from their equilibrium positions, they still move in circles, and do not swing up and down, as Newton believed. It should be noted that Newton was clearly aware of the fallacy of such an assumption, but found it possible to use it for a rough approximate estimate of the speed of wave propagation: in fact, it does not occur in a straight line, but rather in a circle, therefore I assert that time is given to these positions only approximately. Here "time" is the period of oscillations T at each point; wave speed v = %/T, where K is the wavelength. Newton showed that the speed of a wave on water is proportional to -y/K. Later we will see that this is the correct result, and we will find the coefficient of proportionality, which was known to Newton only approximately.
*) We will call cycloids curves described by points lying on the wheel rim, and trochoids - curves described by points between the rim and the axle.
Gerstner's discovery did not go unnoticed. It must be said that he himself continued to be interested in waves and applied his theory to practical calculations of dams and dikes. Soon the laboratory study of waves on water began. This was done by the young Weber brothers.
Elder brother Erist Weber (1795 - 1878) subsequently made important discoveries in anatomy and physiology, especially in the physiology of the nervous system. Wilhelm Weber (1804 - 1891) became a famous physicist and a long-term collaborator of K. Gauss' "control of mathematicians" in physics research. At the suggestion and with the assistance of Gauss, he founded the world's first physical laboratory at the University of Göttingen (1831). Most famous are his works on electricity and magnetism, as well as Weber's electromagnetic theory, which was later superseded by Maxwell's theory. He was one of the first (1846) to introduce the concept of individual particles of electrical matter - "electric masses" and proposed the first model of the atom, in which the atom was likened to a planetary model solar system. Weber also developed the basic theory of Faraday's theory of elementary magnets in matter and invented several physical devices that were very advanced for their time.
Ernst, Wilhelm and their younger brother Eduard Weber became seriously interested in waves. They were real experimenters, and simple observations of the waves, which can be seen "at every step", could not satisfy them. So they made a simple instrument (a Weber tray) which, with various modifications, is still used today for experiments with water waves. Having built a long box with a glass side wall and simple devices for excitation of waves, they carried out extensive observations of various waves, including Gerstner waves, whose theory they thus tested experimentally. They published the results of these observations in 1825 in a book called The Teaching of Waves Based on Experiments. It was the first pilot study, in which waves of various shapes, the speed of their propagation, the relationship between wave length and height, etc. were systematically studied. The methods of observation were very simple, ingenious and quite effective. For example, to determine the shape of the wave surface, they lowered frosted glass
plate. When the wave reaches the middle of the plate, it is quickly pulled out; in this case, the front part of the wave is imprinted quite correctly on the plate. To observe the paths of particles oscillating in a wave, they filled the tray with muddy water from rivers. Saale and observed movements with the naked eye or with a weak microscope. In this way, they determined not only the shape, but also the dimensions of the particle trajectories. So, they found that the trajectories near the surface are close to circles, and when approaching the bottom, they flatten into ellipses; near the bottom, the particles move horizontally. The Webers discovered many interesting properties of waves on water and other liquids.
About the benefits of wave theory
No one seeks his own, but each one seeks the benefit of another.
Apostle Paul
Regardless of this, the development of Lagrange's ideas took place, associated mainly with the names of the French mathematicians Augustin Louis Cauchy (1789 - 1857) and Simon Denis Poisson (1781 - 1840). Our compatriot Mikhail Vasilyevich Ostrogradsky (1801 - 1862) also took part in this work. These famous scientists did a lot for science; numerous equations, theorems and formulas bear their names. Less well known are their works on the mathematical theory of small amplitude waves on the water surface. The theory of such waves can be applied to some storm waves at sea, to the movement of ships, to waves on shallows and near breakwaters, etc. The value of the mathematical theory of such waves for engineering practice is obvious. But at the same time, the mathematical methods developed for solving these practical problems were later applied to solving completely different problems, far from hydromechanics. We will meet again and again with similar examples of the "omnivorousness" of mathematics and the practical benefits of solving mathematical problems, at first glance related to "pure" ("useless") mathematics.
Here it is difficult for the author to refrain from a small digression devoted to one episode associated with the appearance of a single
Ostrogradsky's work on the theory of will. This mathematical work not only brought a distant benefit to science and technology, but also had a direct and important influence on the fate of its author, which does not happen very often. Here is how the outstanding Russian shipbuilder, mathematician and engineer, academician Alexei Nikolaevich Krylov (1863 - 1945) describes this episode. “In 1815 the Parisian Academy of Sciences made the theory of will the subject of the Grand Prix in Mathematics. Cauchy and Poisson took part in the competition. Cauchy's extensive (about 300 pages) memoir was awarded, Poisson's memoir deserved an honorable mention... At the same time (1822) M.V. he was imprisoned in Clichy (a debtor's prison in Paris). Here he wrote "The Theory of Will in a Cylindrical Vessel" and sent his memoir to Cauchy, who not only approved this work and presented it to the Paris Academy of Sciences for publication in its works, but also, not being rich, bought Ostrogradsky out of a debtor's prison and recommended him for the post of teacher of mathematics in one of the lyceums in Paris. A number of Ostrogradsky's mathematical works drew the attention of the St. Petersburg Academy of Sciences to him, and in 1828 he was elected to its adjuncts, and then to ordinary academics, having only a certificate of a student at Kharkov University, who was dismissed without completing the course.
We add to this that Ostrogradsky was born into a poor family of Ukrainian nobles, at the age of 16 he entered the Faculty of Physics and Mathematics of Kharkov University at the behest of his father, contrary to his own desires (he wanted to become a military man), but his outstanding abilities in mathematics soon showed up. In 1820, he passed the exams for a candidate with honors, but the Minister of Public Education and Spiritual Affairs, Kiyaz A.N. Golitsyn, not only refused to award him the degree of candidate, but also deprived him of the previously issued university diploma. The basis was his accusations of "godlessness and free-thinking", that he "did not visit not only
lectures on philosophy, knowledge of God and Christian doctrine. As a result, Ostrogradsky left for Paris, where he diligently attended the lectures of Laplace, Cauchy, Poisson, Fourier, Ampère, and other prominent scientists. Subsequently, Ostrogradsky became a corresponding member of the Paris Academy of Sciences, a member of the Turin,
Roman and American Academies, etc. In 1828, Ostrogradsky returned to Russia, to St. Petersburg, where, on the personal orders of Nicholas I, he was taken under secret police supervision *). This circumstance did not, however, hinder the career of Ostrogradsky, who gradually rose to a very high position.
The work on waves mentioned by A. N. Krylov was published in the Proceedings of the Paris Academy of Sciences in 1826. It is devoted to waves of small amplitude, i.e., the problem that Cauchy and Poissois worked on. Ostrogradskii did not return to the study of waves again. In addition to purely mathematical works, his research on Hamiltonian mechanics is known, one of the first works on the study of the influence of the nonlinear force of friction on the motion of projectiles in the air (this problem was posed as early as
*) Emperor Nicholas I generally treated scientists with distrust, considering all of them, not without reason, freethinkers.
Euler). Ostrogradsky was one of the first who realized the need to study nonlinear oscillations and found an ingenious way to take into account approximately small nonlinearities in pendulum oscillations (the Poisson problem). Unfortunately, he did not complete many of his scientific undertakings - too much effort had to be devoted to pedagogical work, paving the way for new generations of scientists. For this alone, we should be grateful to him, as well as to other Russian scientists of the beginning of the last century, who through hard work created the foundation for the future development of science in our country.
Let us return, however, to our conversation about the benefits of waves. We can give a remarkable example of applying the ideas of wave theory to a completely different range of phenomena. We are talking about Faraday's hypothesis about the wave nature of the process of propagation of electrical and magnetic interactions.
Faraday became a famous scientist during his lifetime, and many studies and popular books have been written about him and his work. However, few people even today know that Faraday was seriously interested in waves on the water. Not knowing the mathematical methods known to Cauchy, Poisson and Ostrogradsky, he understood very clearly and deeply the basic ideas of the theory of waves on water. Thinking about the propagation of electric and magnetic fields in space, he tried to imagine this process by analogy with the propagation of waves on water. This analogy, apparently, led him to the hypothesis about the finiteness of the propagation velocity of electrical and magnetic interactions and about the wave nature of this process. On March 12, 1832, he wrote down these thoughts in a special letter: "New views, which are now to be kept in a sealed envelope in the archives of the Royal Society." The ideas expressed in the letter were far ahead of their time; in fact, the idea of electromagnetic waves was formulated here for the first time. This letter was buried in the archives of the Royal Society, it was discovered only in 1938 by Eidimo, and Faraday himself forgot about it (he gradually developed a serious illness associated with memory loss). He outlined the main ideas of the letter later in the work of 1846.
Of course, today it is impossible to accurately reconstruct Faraday's train of thought. But his reflections and experiments on the waves on the water, shortly before compiling this remarkable letter, are reflected in a work published by him in 1831. It is devoted to the study of small ripples on the surface of water, i.e., the so-called "capillary" waves*) (more about them will be discussed in Chapter 5). For their study, he came up with a witty and, as always, very simple device. Subsequently, Faraday's method was used by Russell, who observed other subtle, but beautiful and interesting phenomena with capillary waves. The experiments of Faraday and Russell are described in § 354 - 356 of Rayleigh's book (John William Stratt, 1842 - 1919) "The Theory of Sound", which was first published in 1877, but is still not outdated and can bring great pleasure to the reader (there is a Russian translation). Rayleigh not only did a lot for the theory of oscillations and waves, but he was also one of the first to recognize and appreciate the solitary wave.
About the main events of the era
The improvement of the sciences should not be expected from the ability or agility of any individual, but from the consistent activity of many generations succeeding each other.
F. Bacon
Meanwhile, it is time for us to finish a somewhat protracted historical excursion, although the picture of science of that time turned out to be, perhaps, too one-sided. In order to somehow correct this, let us briefly recall the events of those years that historians of science rightly consider the most important. As already mentioned, all the basic laws and equations of mechanics were formulated in 1834 in the very form in which we use them today. By the middle of the century, the basic equations describing the motions of fluids and elastic bodies (hydrodynamics and the theory of elasticity) were written and began to be studied in detail. As we have seen, waves in liquids and in elastic bodies have been of interest to many scientists. Physicists, however, were much more fascinated at this time by light waves.
*) These waves are related to the surface tension forces of water. The same forces cause water to rise in the thinnest, hair-thin tubes ( latin word capillus and means hair).
In the first quarter of the century, mainly thanks to the talent and energy of Thomas Young (1773 - 1829), Augustin Jean Fresnel (1788 - 1827) and Dominique Francois Arago (1786 - 1853), the wave theory of light won. The victory was not easy, because among the numerous opponents of the wave theory were such prominent scientists as Laplace and Poisson. A critical experiment that finally approved the wave theory was made by Arago at a meeting of the commission of the Paris Academy of Sciences, which discussed Fresnel's work on the diffraction of light submitted for competition. In the report of the commission, this is described as follows: “One of the members of our commission, Monsieur Poisson, deduced from the integrals reported by the author that amazing result that the center of the shadow from a large opaque screen should be as illuminated as if the screen had not existed ... This consequence was verified by direct experience and observation fully confirmed these calculations.
This happened in 1819, and the following year, Oersted's already mentioned discovery caused a sensation. The publication of Oersted's work "Experiments relating to the action of an electric conflict on a magnetic needle" gave rise to an avalanche of experiments on electromagnetism. It is generally accepted that Ampère made the greatest contribution to this work. Oersted's work was published in Copenhagen at the end of July, at the beginning of September Arago announces this discovery in Paris, and in October the well-known Biot-Savart-Laplace law appears. Since the end of September, Ampere has been performing almost weekly (!) with reports of new results. The results of this pre-Faraday era in electromagnetism are summed up in Ampère's book "The Theory of Electrodynamic Phenomena Derived Exclusively from Experience".
Notice how quickly news of events that aroused general interest spread at that time, although the means of communication were less perfect than today (the idea of telegraph communication was put forward by Ampère in 1829, and it was not until 1844 that the first telegraph began to work in North America commercial telegraph line). The results of Faraday's experiments quickly became widely known. This, however, cannot be said about the spread of Faraday's theoretical ideas that explained his experiments (the concept of lines of force, the electrotonic state, i.e., the electromagnetic field)
The first to appreciate the depth of Faraday's ideas was Maxwell, who was able to find a suitable mathematical language for them.
But this happened already in the middle of the century. The reader may ask why the ideas of Faraday and Ampère were perceived so differently. The point, apparently, is that Ampère's electrodynamics had already matured, "was in the air." Without detracting from the great merits of Ampère, who was the first to give these ideas an exact mathematical form, it must nevertheless be emphasized that Faraday's ideas were much deeper and revolutionary. Oii did not "rush in the air", but were born by the creative power of the thoughts and fantasies of their author. The fact that they were not dressed in mathematical clothes made it difficult to perceive them. If Maxwell had not appeared, Faraday's ideas might have been forgotten for a long time.
The third most important trend in physics in the first half of the last century is the beginning of the development of the theory of heat. The first steps in the theory of thermal phenomena, of course, were connected with the operation of steam engines, and general theoretical ideas were difficult to form and penetrated into science slowly. The remarkable work of Sadi Carnot (1796 - 1832) "Reflections on the driving force of fire and on machines capable of developing this force", published in 1824, went completely unnoticed. It was remembered only thanks to the work of Clapeyron, which appeared in 1834, but the creation of a modern theory of heat (thermodynamics) is a matter of the second half of the century.
Two works are closely related to the questions of interest to us. One of them is the famous book by the outstanding mathematician, physicist and Egyptologist *) Jean Baptiste Joseph Fourier (1768 - 1830) "The Analytical Theory of Heat" (1822), devoted to solving the problem of heat propagation; in it, the method of decomposing functions into sinusoidal components (Fourier expansion) was developed in detail and applied to the solution of physical problems. The birth of mathematical physics as an independent science is usually counted from this work. Its significance for the theory of oscillatory and wave processes is enormous - for more than a century, the main method for studying wave processes has been the decomposition of complex waves into simple sinusoidal waves.
*) After the Napoleonic campaign in Egypt, he compiled a "Description of Egypt" and collected a small but valuable collection of Egyptian antiquities. Fourier directed the first steps of the young Jaya-Fraisois Champolloia, a brilliant decipherer of hieroglyphic writing, the founder of Egyptology. Thomas Jung was also interested in deciphering hieroglyphs, not without success. After studying physics, this was perhaps his main hobby.
(harmonic) waves, or "harmonics" (from "harmony" in music).
Another work is the report of the twenty-six-year-old I Elmholtz "On the Conservation of Force", made in 1847 at a meeting of the Physical Society founded by him in Berlin. Herman Ludwig Ferdinand Helmholtz (1821 - 1894) is rightfully considered one of the greatest natural scientists, and some historians of science put this work of his on a par with the most outstanding works of scientists who laid the foundations of the natural sciences. It deals with the most general formulation of the principle of conservation of energy (then called it “force”) for mechanical, thermal, electrical (“galvanic”) and magnetic phenomena, including processes in an “organized being”. It is especially interesting for us that here Helmholtz first noted the oscillatory nature of the discharge of a Leyden jar and wrote an equation from which W. Thomson soon derived a formula for the period of electromagnetic oscillations in an oscillatory circuit.
In this small work, one can see hints of Helmholtz's future remarkable research. Even a simple enumeration of his achievements in physics, hydromechanics, mathematics, anatomy, physiology and psychophysiology would lead us very far away from the main topic of our story. Let us mention only the theory of vortices in a liquid, the theory of the origin of sea waves, and the first determination of the speed of propagation of an impulse in a nerve. All these theories, as we will soon see, are most directly related to modern research on solitons. Of his other ideas, it is necessary to mention for the first time expressed by him in a lecture on the physical views of Faraday (1881), the idea of the existence of an elementary ("smallest possible") electric charge ("electric atoms"). The electron was discovered experimentally only sixteen years later.
Both described works were theoretical, they formed the foundation of mathematical and theoretical physics. The final development of these sciences is undoubtedly associated with the work of Maxwell, and in the first half of the century, purely theoretical approach to physical phenomena was, in general, alien to most
puppies. Physics was considered a purely "experimental" science, and even in the titles of the works, the main words were "experiment", "based on experiments", "derived from experiments". It is interesting that Helmholtz's work, which even today can be considered a model of depth and clarity of exposition, was not accepted by a physics journal as theoretical and too large in volume and was later published as a separate pamphlet. Shortly before his death, Helmholtz spoke of the history of the creation of his most famous work:
“Young people are most willing to take on the most profound tasks at once, and so I was occupied with the question of the mysterious essence of vital force ... I found that ... the theory of vital force ... attributes to every living body the properties of a “perpetual motion machine” ... Looking through the writings of Daniel Bernoulli, D'Alembert and other mathematicians of the last century ... I came across the question: “what relations should exist between the various forces of nature, if we accept that a “perpetual motion machine” is impossible at all and whether all these relations are actually fulfilled. ..” I intended only to give a critical assessment and systematics of the facts in the interests of physiologists. It would not be a surprise to me if in the end knowledgeable people said to me: “Yes, all this is well known. What does this young medic want by going into such detail about these things?” To my surprise, the physicists with whom I came into contact took a completely different view of the matter. They were inclined to reject the justice of the law; in the midst of the zealous struggle they had with Hegel's natural philosophy, and my work was considered fantastic speculation. Only mathematician Jacobi recognized the connection between my reasoning and the thoughts of mathematicians of the last century, became interested in my experience and protected me from misunderstandings.
These words clearly characterize the mindset and interests of many scientists of that era. There is, of course, a regularity and even a necessity in such resistance of the scientific community to new ideas. So let's not rush to condemn Laplace, who did not understand Fresnel, Weber, who did not recognize Faraday's ideas, or Kelvin, who opposed the recognition of Maxwell's theory, but rather let us ask ourselves whether it is easy for us to assimilate new ideas, unlike everything we are used to. . We recognize that some conservatism is inherent in our human nature, and hence in the science that people do. It is said that a certain "healthy conservatism" is even necessary for the development of science, as it prevents the spread of empty fantasies. However, this is by no means comforting when one recalls the fates of geniuses who looked into the future, but were not understood and not recognized by their era.
Your age, marveling at you, did not comprehend the prophecies
And mixed insane reproaches with flattery.
V. Bryusov
Perhaps the most striking examples of such a conflict with the era in the time of interest to us (about 1830) we see in the development of mathematics. The face of this science was then determined, probably, by Gauss and Cauchy, who, together with others, completed the construction of a great building mathematical analysis without which modern science is simply unthinkable. But we cannot forget that at the same time, not appreciated by contemporaries, the young Abel (1802 - 1829) and Galois (1811 - 1832) died, which from 1826 to 1840. Lobachevsky (1792 - 1856) and Bolyai (1802 - 1860) published their works on non-Euclidean geometry, who did not live to see their ideas recognized. The reasons for this tragic misunderstanding are deep and manifold. We cannot delve into them, but we will give just one more example that is important for our story.
As we will see later, the fate of our hero, the soliton, is closely connected with computers. Moreover, history presents us with a striking coincidence. In August 1834, while Russell was observing a solitary wave, the English mathematician, economist, and inventor Charles Babbege (1792 - 1871) completed the development of the basic principles of his "analytical" machine, which later formed the basis of modern digital computers. Babbage's ideas were far ahead of their time. It took more than a hundred years to realize his dream of building and using such machines. It is difficult to blame Babbage's contemporaries for this. Many understood the need for computers, but technology, science and society were not yet ripe for the implementation of his bold projects. The Prime Minister of England, Sir Robert Peel, who had to decide the fate of the financing of the project presented by Babbage to the government, was not ignorant (he graduated from Oxford first in mathematics and classics). He held a formally thorough discussion of the project, but as a result he came to the conclusion that the creation of a universal computer was not among the priorities of the British government. It was not until 1944 that the first automatic digital machines appeared, and an article entitled "Babbage's dream came true" appeared in the English journal Nature.
Science and society
A team of scientists and writers... is always ahead in all iabegas of enlightenment, in all attacks of education. They should not cowardly be indignant at the fact that they are forever destined to endure the first shots and all hardships, all dangers.
A. S. Pushkin
Of course, both the successes of science and its failures are connected with the historical conditions of the development of society, on which we cannot detain the reader's attention. It is no coincidence that at that time there was such a pressure of new ideas that science and society did not have time to master them.
The development of science in different countries followed different paths.
In France scientific life was united and organized by the Academy to such an extent that work not noticed and supported by the Academy, or at least by well-known academicians, had little chance of being of interest to scientists. But the works that fell into the field of view of the Academy were supported and developed. This sometimes caused protests and indignation on the part of young scientists. In an article dedicated to the memory of Abel, his friend Szegi wrote: “Even in the case of Abel and Jacobi, the favor of the Academy did not mean recognizing the undoubted merits of these young scientists, but rather the desire to encourage the study of certain problems relating to a strictly defined range of questions, beyond which, in the opinion Academy, there can be no progress in science and no valuable discoveries can be made ... We will say something completely different: young scientists, do not listen to anyone but your own inner voice. Read and meditate on the works of geniuses, but never become dispossessed students.
opinion... Freedom of views and objectivity of judgment - this should be your motto. (Perhaps "not listening to anyone" is a polemical exaggeration, the "inner voice" is not always right.)
In many small states that were on the territory of the future German Empire (it was only by 1834 that customs were closed between most of these states), scientific life was concentrated in numerous universities, most of which also carried out research work. It was there at that time that schools of scientists began to take shape and a large number of scientific journals were published, which gradually became the main means of communication between scientists, not subject to space and time. Their pattern is followed by modern scientific journals.
In the British Isles there was no academy of the French type, promoting the achievements recognized by it, nor such scientific schools like in Germany. Most English scientists worked alone*). These loners managed to pave completely new paths in science, but their work often remained completely unknown, especially when they were not sent to a journal, but were only reported at meetings of the Royal Society. The life and discoveries of the eccentric nobleman and brilliant scientist, Lord Henry Cavendish (1731 - 1810), who worked all alone in his own laboratory and published only two works (the rest, containing discoveries rediscovered by others only decades later, were found and published by Maxwell), especially vividly illustrate these features of science in England at the turn of the 18th - 19th centuries. Such trends in scientific work persisted in England for quite a long time. For example, the already mentioned Lord Rayleigh also worked as an amateur, he performed most of his experiments in his estate. This "amateur", in addition to a book on the theory of sound, wrote
*) Don't take it too literally. Any scientist needs constant communication with other scientists. In England, the center of such communication was the Royal Society, which also had considerable funds to finance scientific research.
more than four hundred works! Maxwell also worked alone in his family nest for several years.
As a result, as the English historian of science wrote about this time, “the largest number of works perfected in form and content that have become classics ... probably belongs to France; the largest number of scientific papers was probably carried out in Germany; but among the new ideas that have fertilized science for a century, the largest share probably belongs to England. The last statement can hardly be attributed to mathematics. If we talk about physics, then this judgment does not seem too far from the truth. Let us also not forget that Russell's contemporary *) was the great Charles Darwin, who was born a year later and died the same year as him.
What is the reason for the success of lone researchers, why were they able to come up with such unexpected ideas that they seemed to many other equally gifted scientists not only wrong, but even almost crazy? If we compare Faraday and Darwin - two great naturalists of the first half of the last century, then their extraordinary independence from the teachings that prevailed at that time, trust in their own vision and reason, great ingenuity in posing questions and the desire to fully understand the unusual that they managed to observe. It is also important that an educated society is not indifferent to scientific research. If there is no understanding, then there is interest, and a circle of admirers and sympathizers usually gathers around the pioneers and innovators. Even Babbage, who was misunderstood and became a misanthrope by the end of his life, had people who loved and appreciated him. Darwin understood and highly appreciated him; an outstanding mathematician, Byron's daughter, Lady
*) Most of the contemporaries mentioned by us were probably familiar with each other. Of course, members of the Royal Society met at meetings, but, in addition, they also maintained personal contacts. For example, it is known that Charles Darwin visited Charles Babbage, who from his student years was friends with John Herschel, who knew John Russell closely, etc.
Ada Augusta Lovelace. Babbage was also appreciated by Faraday and other prominent people of his time.
The social significance of scientific research has already become clear to many educated people, and this sometimes helped scientists to receive the necessary funds, despite the lack of centralized funding for science. By the end of the first half of XVIII V. The Royal Society and the leading universities had more resources than any of the leading scientific institutions on the Continent. “... A galaxy of outstanding physicists, like Maxwell, Rayleigh, Thomson ... could not have arisen if ... in England at that time there had not been a cultural scientific community that correctly assessed and supported the activities of scientists” (P L. Kapitsa).
END OF CHAPTER AND FRAGMEHTA OF THE BOOK
