Fundamental soliton and its use. shock waves

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 (really, a telling 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 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, the broadband (that is, the ability to transmit many frequencies, and hence useful information) of fiber-optic communication lines is limited by nonlinear 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, an equation of the form can be obtained for the amplitude of the electric field inside the fiber

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 attract attention and make one think about practical application for about 40 years now. 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

1. J. Lem. Introduction to the theory of solitons. Per. from English. M.: Mir, - 1983. -294 p.
2. J. Whitham Linear and non-linear waves. - M.: Mir, 1977. - 624 p.
3. I. R. Shen. Principles of nonlinear optics: Per. from English / Ed. S. A. Akhmanova. - M.: Nauka., 1989. - 560 p.
4. S. A. Bulgakova, A. L. Dmitriev. Nonlinear optical information processing devices// Tutorial. - St. Petersburg: SPbGUITMO, 2009. - 56 p.
5. Werner Alpers et. al. Observation of Internal Waves in the Andaman Sea by ERS SAR // Earthnet Online
6. 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.
7. N. N. Rozanov. World of laser solitons // Nature, 6 (2006). pp. 51-60.
8. O. A. Tatarkina. Some aspects of the design of soliton fiber-optic transmission systems // Fundamental research, 1 (2006), pp. 83-84.

P.S. About diagrams in .

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1. Introduction

Waves in nature

It is well known from a school course in physics that if vibrations are excited at any point in an elastic medium (solid, liquid or gaseous), then they will be transmitted to other places. This transfer of excitations is due to the fact that close parts of the medium are connected to each other. In this case, vibrations excited in one place propagate in space at a certain speed. It is customary to call a wave the process of transferring excitations of a medium (in particular, an oscillatory process) from one point to another.

The nature of the wave propagation mechanism can be different. In the simplest case, the bonds between sections in the medium can be due to elastic forces that arise due to deformations in the medium. In this case, both longitudinal waves, in which the displacements of the particles of the medium are carried out in the direction of wave propagation, and transverse waves, in which the displacements of particles are perpendicular to the propagation of the wave, can propagate in a solid elastic medium. In a liquid or gas, unlike solids, there are no shear resistance forces, so only longitudinal waves can propagate. A well-known example of longitudinal waves in nature are sound waves, which are produced due to the elasticity of air.

Among waves of a different nature, a special place is occupied by electromagnetic waves, the transfer of excitations in which occurs due to fluctuations in electric and magnetic fields. The medium in which electromagnetic waves propagate, as a rule, has a significant impact on the process of wave propagation, however, electromagnetic waves, unlike elastic ones, can propagate even in a vacuum. The connection between different sections in space during the propagation of such waves is due to the fact that a change in the electric field causes the appearance of a magnetic field and vice versa.

With the phenomena of the propagation of electromagnetic waves, we often encounter in our daily lives. These phenomena include radio waves, the use of which in technical applications is well known. In this regard, we can mention the work of radio and television, which is based on the reception of radio waves. Electromagnetic phenomena, only in a different frequency range, also include light, with which we see the objects around us.

A very important and interesting type of waves are waves on the surface of the water. This is one of the common types of waves that everyone has observed since childhood and which is usually demonstrated as part of a school physics course. However, in the words of Richard Feynman, "it is difficult to think of a more unfortunate example for demonstrating waves, because these waves are in no way similar to either sound or light; here all the difficulties that can be in waves have gathered."

If we consider a sufficiently deep pool filled with water, and create some disturbance on its surface, then waves will begin to propagate over the surface of the water. Their occurrence is explained by the fact that the liquid particles that are near the cavity, when creating a disturbance, will tend to fill the cavity, being under the action of gravity. The development of this phenomenon over time will lead to the propagation of waves on the water. The particles of the liquid in such a wave do not move up and down, but approximately in circles, so the waves on the water are neither longitudinal nor transverse. They are like a mixture of both. With depth, the radii of the circles along which fluid particles move decrease until they become equal to zero.

If we analyze the speed of wave propagation on water, it turns out that it depends on its length. The speed of long waves is proportional to the square root of the gravitational acceleration times the wavelength. The reason for the occurrence of such waves is the force of gravity.

For short waves, the restoring force is due to the surface tension force, and therefore the speed of such waves is proportional to the square root of the quotient, the numerator of which is the surface tension coefficient, and the denominator is the product of the wavelength and the density of water. For medium wavelength waves, their propagation velocity depends on the above parameters of the problem. From what has been said, it is clear that waves on water are indeed a rather complex phenomenon.

1.2. The discovery of a solitary wave

Waves on the water have long attracted the attention of researchers. This is due to the fact that they are a well-known phenomenon in nature and, in addition, accompany the movement of ships through the water.

A curious wave on the water was observed by the Scottish scientist John Scott Russell in 1834. He was engaged in research on the movement of a barge along the canal, which was pulled by a pair of horses. Suddenly, the barge stopped, but the mass of water that the barge set in motion did not stop, but gathered at the bow of the ship, and then broke away from it. Further, this mass of water rolled along the canal at high speed in the form of a solitary elevation, without changing its shape and without slowing down.

Throughout his life, Russell repeatedly returned to observing this wave, because he believed that the solitary wave he discovered plays an important role in many phenomena in nature. He established some properties of this wave. First noticed that she was moving with constant speed and without changing shape. Secondly, I found the dependence of the speed FROM this wave from the depth of the channel h and wave height a:

where g - free fall acceleration, and a < h . Thirdly, Russell discovered that it is possible for one big wave to break up into several waves. Fourth, he noted that only elevation waves are observed in experiments. Once he also noticed that the solitary waves he discovered pass through each other. without any change, as well as small waves formed on the surface of the water. However, he did not pay much attention to the last very important property.

Russell's work, published in 1844 as A Report on Waves, provoked a cautious reaction among scientists. On the Continent, she was not noticed at all, and in England itself, G.R. Airey and J.G. Stock. Airy criticized the results of the experiments that Russell observed. He noted that Russell's theory of long waves in shallow water did not lead to conclusions, and argued that long waves could not maintain an unchanged shape. And ultimately questioned the validity of Russell's observations. One of the founders of modern hydrodynamics, George Gabriel Stoke, also disagreed with Russell's observations and was critical of the existence of a solitary wave.

After such a negative attitude towards the discovery of a solitary wave, for a long time they simply did not remember about it. A certain clarity in Russell's observations was introduced by J. Boussinesq (1872) and J.W. Rayleigh (1876), who independently found an analytical formula for the elevation of a free surface on water in the form of a square of hyperbolic secant and calculated the propagation velocity of a solitary wave on water.

Later, Russell's experiments were repeated by other researchers and received confirmation.

1.3. Linear and non-linear waves

Partial differential equations are often used as mathematical models for describing the propagation of waves in various media. These are equations that contain as unknowns the derivatives of the characteristics of the phenomenon under consideration. Moreover, since the characteristic (for example, the density of air during sound propagation) depends on the distance to the source and on time, then not one, but two (and sometimes more) derivatives are used in the equation. The simple wave equation has the form

u tt = c 2 u xx (1.1)

Wave characteristic and in this equation depends on the spatial coordinate X and time t , and the indexes of the variable and denote the second derivative of and by time ( u tt) and the second derivative of and by variable x (u xx ). Equation (1) describes a plane one-dimensional wave, which can be analogous to a wave in a string. In this equation, as and we can take the density of air, if we are talking, for example, about a sound wave in air. If we consider electromagnetic waves, then under and should be understood as the strength of the electric or magnetic field.

The solution of the wave equation (1), which was first obtained by J. D "Alembert in 1748, has the form

u(x,t)=f(x-ct)+g(x+ct) (1.2)

Here the functions f and g are found from the initial conditions for and. Equation (1.1) contains the second derivative of and on t , therefore, two initial conditions should be specified for it: the value and at t = 0 and derivative and, at t = 0.

Wave equation (1.1) has a very important property, the essence of which is as follows. It turned out that if we take any two solutions of this equation, then their sum will again be a solution to the same equation. This property reflects the principle of superposition of solutions to equation (1.1) and corresponds to the linearity of the phenomenon it describes. For nonlinear models, this property is not satisfied, which leads to significant differences in the course of processes in the corresponding models. In particular, from the expression for the velocity of a solitary wave observed by Russell, it follows that its value depends on the amplitude, while for the wave described by equation (1.1) there is no such dependence.

By direct substitution into equation (1.1), one can verify that the dependence

u(x,t)=a cos(kx-  t) (1.3)

wherein a,k and  - permanent, at  =± k is a solution to equation (1). In this decision a - amplitude, k is the wave number, and  - frequency. The above solution is a monochromatic wave transported in a medium with a phase velocity

c p = (1.4)

In practice, it is difficult to create a monochromatic wave, and usually one deals with a train (packet) of waves, in which each wave propagates at its own speed, and the packet propagation velocity is characterized by the group velocity

C g = , (1.5)

defined through the derivative of the frequency  by wave number k .

It is not always easy to determine which (linear or non-linear) model the researcher is dealing with, but when the mathematical model is formulated, the solution of this issue is simplified and the implementation of the principle of superposition of solutions can be verified.

Returning to water waves, we note that they can be analyzed using the well-known equations of hydrodynamics, which are known to be non-linear. Therefore, the waves on the water in general case are non-linear. Only in the limiting case of small amplitudes can these waves be considered linear.

Note that the propagation of sound is not in all cases described by a linear equation. Even Russell, when substantiating his observations on a solitary wave, noted that the sound from a cannon shot propagates in the air faster than the command to fire this shot. This is explained by the fact that the propagation of powerful sound is no longer described by the wave equation, but by the equations of gas dynamics.

1. Korteweg - de Vries equation

The final clarity in the problem that arose after Russell's experiments on a solitary wave came after the work of the Danish scientists D.D. Korteweg and G. de Vries, who tried to understand the essence of Russell's observations. Generalizing the Rayleigh method, these scientists in 1895 derived an equation to describe long waves on water. Korteweg and de Vries, using the equations of hydrodynamics, considered the deviation them,t ) on the equilibrium position of the water surface in the absence of vortices and at a constant density of water. The initial approximations they made were natural. They also assumed that two conditions for the dimensionless parameters are satisfied during wave propagation

= <<1,  = (2.1)

Here a - wave amplitude, h - the depth of the pool in which the waves are considered, l- wavelength (Fig. 1).

The essence of the approximations was that the amplitude of the considered waves was much smaller than

Rice. 1. A solitary wave propagating through a channel and its parameters

the depth of the basin, but at the same time the wavelength was much greater than the depth of the basin. Thus, Korteweg and de Vries considered long waves.

The equation they obtained is

u t + 6uu x +u xxx = 0. (2.2)

Here u (x,t) - deviation from the equilibrium position of the water surface (waveform) - depends on the coordinate x and time t. Characteristic indexes u mean the corresponding derivatives with respect to t and by x . This equation, like (1), is a partial differential equation. The studied characteristic of him (in this case u ) depends on the spatial coordinate x and time t .

To solve an equation of this type means to find the dependence u from x and t, after substituting which into the equation, we arrive at an identity.

Equation (2.2) has a wave solution known since the end of the last century. It is expressed in terms of a special elliptic function studied by Carl Jacobi, which now bears his name.

Under certain conditions, the Jacobi elliptic function transforms into a hyperbolic secant and the solution has the form

u(x,t)=2k 2 ch -2 (k(x-4k 2 t)+  0 } , (2.3)

where  0 is an arbitrary constant.

Solution (8) of equation (7) is the limiting case of an infinitely large period of the wave. It is this limiting case that is the solitary wave corresponding to Russell's observation in 1834.

Solution (8) of the Korteweg-de Vries equation is a traveling wave. This means that it depends on the coordinate x and time t through a variable  = x - c 0 t . This variable characterizes the position of the coordinate point moving with the speed of the wave c0, that is, it denotes the position of the observer, who is constantly on the crest of the wave. Thus, the Kortewegade-Vries equation, in contrast to the d'Alembert solution (1.2) of the wave solution (1.1), has a wave propagating in only one direction. However, it takes into account the manifestation of more complex effects due to additional terms uu x and u xxx .

In fact, this equation is also approximate, since small parameters (2.1) were used in its derivation  and . If we neglect the influence of these parameters, tending them to zero, we get one of the parts of the d'Alembert solution.

Of course, when deriving the equation for long waves on water, the influence of the parameters e and 6 can be taken into account more accurately, but then an equation will be obtained that contains many more terms than equation (2.2), and with derivatives of a higher order. From what has been said, it follows that the solution of the Korteweg-de Vries equation for describing waves is valid only at a certain distance from the place of wave formation and at a certain time interval. At very large distances, nonlinear waves will no longer be described by the Korteweg-de Vries equation, and a more accurate model will be required to describe the process. The Korteweg-de Vries equation in this sense should be considered as some approximation (mathematical model) corresponding with a certain degree of accuracy to the real process of wave propagation on water.

Using a special approach, one can verify that the principle of superposition of solutions for the Korteweg-de Vries equation does not hold, and therefore this equation is nonlinear and describes nonlinear waves.

2.1. Korteweg-de Vries solitons

At present, it seems strange that Russell's discovery and its subsequent confirmation in the work of Korteweg and de Vries did not receive a noticeable resonance in science. These works were forgotten for almost 70 years. One of the authors of the equation, D.D. Korteweg lived a long life and was a famous scientist. But when in 1945 the scientific community celebrated his 100th anniversary, the work he and de Vries did not even appear on the list of the best publications. The compilers of the list considered this work by Korteweg not worthy of attention. Only a quarter of a century later, it was this work that began to be considered the main scientific achievement of Korteweg.

However, if you think about it, such inattention to Russell's solitary wave becomes understandable. The fact is that, due to its specificity, this discovery has long been considered a rather private fact. Indeed, at that time the physical world seemed linear and the principle of superposition was considered one of the fundamental principles of most physical theories. Therefore, none of the researchers attached serious importance to the discovery of an exotic wave on the water.

The return to the discovery of a solitary wave on the water was somewhat accidental and at first seemed to have nothing to do with it. The culprit of this event was the greatest physicist of our century, Enrico Fermi. In 1952, Fermi asked two young physicists S. Ulam and D. Pasta to solve one of the nonlinear problems on a computer. They had to calculate the vibrations of 64 weights connected to each other by springs, which, when deviated from the equilibrium position by  l acquired a returning force equal to k  l +a( l) 2 . Here k and a- constant coefficients. In this case, the nonlinear additive was assumed to be small compared to the main force k  l. By creating the initial oscillation, the researchers wanted to see how this initial mode would be distributed over all other modes. After carrying out the calculations of this problem on a computer, they did not obtain the expected result, but found that the transfer of energy into two or three modes at the initial stage of the calculation actually occurs, but then a return to the initial state is observed. This paradox associated with the return of the initial oscillation has become known to several mathematicians and physicists. In particular, the American physicists M. Kruskal and N. Zabuski learned about this problem and decided to continue their computational experiments with the model proposed by Fermi.

After calculations and searching for analogies, these scientists found that the equation used by Fermi, Pasta and Ulam, with a decrease in the distance between the weights and with an unlimited increase in their number, goes into the Korteweg-de Vries equation. That is, in essence, the problem proposed by Fermi was reduced to the numerical solution of the Korteweg-de Vries equation, proposed in 1895 to describe a solitary Russell wave. Approximately in the same years, it was shown that the Korteweg-de Vries equation is also used to describe ion-acoustic waves in plasma. Then it became clear that this equation is found in many areas of physics and, therefore, the solitary wave, which is described by this equation, is a widespread phenomenon.

Continuing the computational experiments to model the propagation of such waves, Kruskal and Zabusky considered their collision. Let us dwell in more detail on the discussion of this remarkable fact. Let there be two solitary waves described by the Korteweg-de Vries equation, which differ in amplitude and move one after the other in the same direction (Fig. 2). It follows from the formula for solitary waves (8) that the higher the velocity of such waves, the greater their amplitude, and the peak width decreases with increasing amplitude. Thus, high solitary waves move faster. A wave with a larger amplitude will overtake a wave with a smaller amplitude moving ahead. Then, for some time, the two waves will move together as a whole, interacting with each other, and then they will separate. A remarkable property of these waves is that after their interaction, the form and

Rice. 2. Two solitons described by the Korteweg-de Vries equation,

before interaction (top) and after (bottom)

the speed of these waves is restored. Both waves after the collision are only displaced by a certain distance compared to how they would move without interaction.

The process, in which the shape and speed are preserved after the interaction of waves, resembles an elastic collision of two particles. Therefore, Kruskal and Zabuski called such solitary waves solitons (from the English solitary - solitary). This special name for solitary waves, consonant with the electron, proton and many other elementary particles, is currently generally accepted.

Solitary waves, which were discovered by Russell, indeed behave like particles. A large wave does not pass through a small one during their interaction. When solitary waves touch, the large wave slows down and decreases, and the wave that was small, on the contrary, accelerates and grows. And when the small wave grows to the size of a large one, and the large one decreases to the size of a small one, the solitons separate and the larger one moves forward. Thus, solitons behave like elastic tennis balls.

Let's give a definition of a soliton. Soliton called a non-linear solitary wave, which retains its shape and speed during its own movement and collision with similar solitary waves, that is, it is a stable formation. The only result of the interaction of solitons can be some phase shift.

The discoveries related to the Korteweg-de Vries equation did not end with the discovery of the soliton. The next important step related to this remarkable equation was the creation of a new method for solving non-linear partial differential equations. It is well known that finding solutions to nonlinear equations is very difficult. Until the 1960s, it was believed that such equations could only have certain particular solutions that satisfy specially given initial conditions. However, the Korteweg-de Vries equation also found itself in an exceptional position in this case.

In 1967, American physicists K.S. Gardner, J.M. Green, M. Kruskal and R. Miura have shown that the solution of the Korteweg-de Vries equation can in principle be obtained for all initial conditions that vanish in a certain way as the coordinate tends to infinity. They used the transformation of the Korteweg-de Vries equation to a system of two equations, now called the Lax pair (after the American mathematician Peter Lax, who made a great contribution to the development of the theory of solitons), and discovered a new method for solving a number of very important non-linear partial differential equations. This method is called the method of the inverse scattering problem, since it essentially uses the solution of the problem of quantum mechanics about the reconstruction of the potential from scattering data.

2.2. Group soliton

Above, we said that in practice the waves, as a rule, propagate in groups. Similar groups of waves on the water people have observed since time immemorial. T. Benjamin and J. Feyer managed to answer the question of why "flocks" of waves are so typical for waves on water, only in 1967. By theoretical calculations, they showed that a simple periodic wave in deep water is unstable (now this phenomenon is called the Benjamin-Fejér instability), and therefore the waves on the water are divided into groups due to instability. The equation that describes the propagation of wave groups on water was obtained by V.E. Zakharov in 1968. By that time, this equation was already known in physics and was called the nonlinear Schrödinger equation. In 1971, V.E. Zakharov and A.B. Shabat showed that this nonlinear equation also has solutions in the form of solitons, moreover, the nonlinear Schrödinger equation, as well as the Korteweg-de Vries equation, can be integrated by the method of the inverse scattering problem. The solitons of the nonlinear Schrödinger equation differ from the Korteweg-de Vries solitons discussed above in that they correspond to the shape of the wave group envelope. Outwardly, they resemble modulated radio waves. These solitons are called group solitons and sometimes envelope solitons. This name reflects the persistence in the interaction of the wave packet envelope (analogous to the dashed line shown in Fig. 3), although the waves themselves under the envelope move at a speed different from the group speed. In this case, the shape of the envelope is described

Rice. 3. An example of a group soliton (dashed line)

addiction

a(x,t)=a 0 ch -1 (
)

where aa - amplitude, and l is half the size of the soliton. Usually, there are from 14 to 20 waves under the envelope of a soliton, with the middle wave being the largest. Related to this is the well-known fact that the highest wave in a group on water is between the seventh and tenth (the ninth wave). If a larger number of waves has formed in a group of waves, then it will break up into several groups.

The nonlinear Schrödinger equation, like the Korteweg-de Vries equation, is also widely used in the description of waves in various fields of physics. This equation was proposed in 1926 by the outstanding Austrian physicist E. Schrödinger to analyze the fundamental properties of quantum systems and was originally used to describe the interaction of intraatomic particles. The generalized or nonlinear Schrödinger equation describes a set of phenomena in the physics of wave processes. For example, it is used to describe the effect of self-focusing when a powerful laser beam acts on a nonlinear dielectric medium and to describe the propagation of nonlinear waves in a plasma.

3. Statement of the problem

3.1. Description of the model. At present, there is a significantly growing interest in the study of nonlinear wave processes in various fields of physics (for example, in optics, plasma physics, radiophysics, hydrodynamics, etc.). To study waves of small but finite amplitude in dispersive media, the Korteweg-de Vries (KdV) equation is often used as a model equation:

ut + AIX +  andxxx = 0 (3.1)

The KdV equation was used to describe magnetosonic waves propagating strictly across the magnetic field or at angles close to .

The main assumptions that are made when deriving the equation are: 1) small but finite amplitude, 2) the wavelength is large compared to the dispersion length.

Compensating the effect of nonlinearity, dispersion makes it possible to form in a dispersive medium stationary waves of finite amplitude - solitary and periodic. The solitary waves for the KdV equation came to be called solitons after the work. Periodic waves are called cnoidal waves. The corresponding formulas for their description are given in.

3.2. Formulation of a differential problem. In this paper, we study the numerical solution of the Cauchy problem for the Korteweg-de Vries equation with periodic conditions in space in a rectangle Q T ={(t , x ):0< t < T , x [0, l ].

ut + AIX +  andxxx = 0 (3.2)

u(x,t)| x=0 =u(x,t)| x=l (3.3)

with initial condition

u(x,t)| t=0 =u 0 (x) (3.4)

4. Properties of the Korteweg - de Vries equation

4.1. A brief review of the results on the KdV equation. The Cauchy problem for the KdV equation under various assumptions about u 0 (X) considered in many works. The problem of the existence and uniqueness of a solution with periodicity conditions as boundary conditions was solved in this work using the finite difference method. Later, under less strong assumptions, the existence and uniqueness were proved in the article in the space L  (0,T ,H s (R ​​1)), where s>3/2, and in the case of a periodic problem - in the space L  (0 ,T ,H  (C )) where C is a circle of length equal to the period, in Russian these results are presented in the book.

The case when no smoothness of the initial function is assumed u 0  L 2 (R 1 ) , considered in the work. There, the concept of a generalized solution of problem (3.2),(3.4) is introduced, the existence of a generalized solution is established and(t ,X)  L  (0, T , L 2 (R 1 )) in the case of an arbitrary initial function u 0  L 2 (R 1 ) ; wherein and(t ,X) L 2 (0,T;H -1 (- r , r )) for anyone r>0, and if for some  > 0 (x  u 0 2 (x ))  L 1 (0,+  ) , then

(4.1)

Using the inversion of the linear part of the equation by using the fundamental solution G (t,x) corresponding linear operator
, a well-posedness class of problem (3.2),(1.4) is introduced, and theorems of uniqueness and continuous dependence of solutions of this problem on the initial data are established. Questions of regularity of generalized solutions are also investigated. One of the main results is a sufficient condition for the existence of a H ¨older continuous for t > 0 derivative
in terms of the existence of moments for the initial function, for any k and l .

The Cauchy problem for the KdV equation was also studied by the inverse scattering method proposed in . Using this method, results were obtained on the existence and smoothness of solutions for sufficiently rapidly decreasing initial functions, and in particular, a result was established on the solvability of problem (3.2),(3.4) in the space C  (O, T; S(R 1 )) .

Most full review modern results on the KdV equation can be found in .

4.2. Conservation laws for the KdV equation. As is known, for the KdV equation, there are an infinite number of conservation lawsniya. This paper provides a rigorous proof of this fact.In papers, various conservation laws have been applied to beforeproofs of nonlocal existence theorems for a solution to problem (3.2),(3.4) from the corresponding spaces.

Let us demonstrate the derivation of the first three conservation laws for Kosha's dachas on R 1 and a periodic task.

To obtain the first conservation law, it suffices tocalculate equations (3.2) with respect to the space variable. Semi chim:

hence the first conservation law follows:

Here asa and b+  and -  stand out for the Cauchy problem and boundaries of the main period for the periodic problem. That's whythe second and third terms go to 0.

(4.2)

To derive the second conservation law, one should multiply the equation(3.2) on 2 u (t,x) and integrate over the spacechange. Then, using the formula for integration by parts, the floor chim:

but due to the "boundary" conditions, all terms except the first one again are shrinking

Thus, the second integral conservation law has the form:

(4.3)

To derive the third conservation law, we need to multiply our equation (3.2) by (and 2 + 2  and xx ), thus we get:

After applying integration by parts several times, the third and fourth integrals cancel out. Second and third termswe disappear because of the boundary conditions. So from the firstintegral we get:

which is equivalent to

And this is the third conservation law for equation (3.2).Under the physical meaning of the first two integral laws withstorage in some models it is possible to understand the conservation laws momentum and energy, for the third and subsequent conservation laws it is already more difficult to characterize the physical meaning, but from the point of view of mathematics, these laws provide additional information about the solution, which is then used to prove existence and uniqueness theorems for the solution, study its properties and derive a priori estimates.

5. Difference schemes for solving the KdV equation

3.1. Notation and formulation of the difference problem. In the area of ={( x , t ):0  x  l ,0  t  T } we introduce in the usual wayuniform grids, where

We introduce a linear space h grid functions defined on a grid
with values ​​at grid nodesy i = y h ( x i ). Prev it is assumed that the periodicity conditions are satisfiedy 0 = y N . Except moreover, we formally assumey i + N = y i for i  1 .

We introduce the scalar product in space  h

(5.1)

We endow the linear space P/r with the norm:

Because into space h includes periodic functions, thenthis dot product is equivalent to the dot product niyu:

We will construct difference schemes for equation (3.2) on a grid with periodic boundary conditions. We need the notation for difference approximations. Let's introduce them.

We use standard notation to solve the equation on the next (n-m) time layer, that is

Let us introduce notation for difference approximations of derivatives.For the first time derivative:

Similarly for the first derivative with respect to space:

Now we introduce the notation for the second derivatives:

The third spatial derivative will be approximated as follows:

We also need an approximation for 2 , which we will denote letter Q and enter as follows:

(5.2)

To write the equation on semi-integral layers, we will usebalanced approximation, i.e.

except for the approximationat 2 on the whole floor. Let's bringone of the possible approximationsat 2 on the whole floor:

Comment 2. It should be noted that for 1 equality holds:

Definition 1. Following the difference scheme for the KdV equationwill be called conservative if it has a gridanalogous to the first integral conservation law, it is true

Definition 2. Following the difference scheme for the KdV equation, we will callL 2 -conservative if there is a grid for itanalogous to the second integral conservation law, it is trueth for the differential problem.

5.2. Explicit difference schemes (review).When building timesdifference schemes, we will focus on the simplest differencescheme from the paper for the linearized KdV equation, whichwhich preserves the properties of the KdV equation itself in the sense of the first twoconservation laws.

(5.3)

Let us now examine the scheme (5.4) for its conservative properties. youthe fulfillment of the first conservation law is obvious. Simple enoughmultiply this equation scalarly by 1. Then the second and third wordsthe schemes (5.4) will give 0, and the first will remain:

(5.4)

This is a grid analogue of the first conservation law.

To derive the second conservation law, we scalarly multiply the equation(5.3) on 2  y. Coming to energy identity

(5.5)

The presence of a negative imbalance indicates not only unfulfilledthe corresponding conservation law, but also casts doubt on the general stability of the scheme in the weakest normL 2 (). )- We show that schemes of the family (3.18) areabsolutely unstable in the normL 2 ().

At the moment, the most popular schemes for the equationKdV are considered three-layer circuits due to their simplicity, accuracy andease of implementation.

(5.6)

The same scheme can be represented as an explicit formula

The simplest three-layer circuit is the following circuit:

This scheme was used to obtain the first numerical solutions of KdV. This scheme approximates a differential problem with order O ( 2 + h 2 ). According to , the scheme is stableviable under the condition (for small b):

Let's take a look at a few more diagrams. Three-layer explicit scheme with ordercom approximationO (  2 + h 4 ) :

The third spatial derivative is approximated by sevendot pattern, and the first one is based on five points. According to ,this scheme is stable under the condition (for smallh ):

It is easy to see that for this scheme with a higher order of approximation, the stability condition is more stringent.

We propose the following explicit difference scheme withapproximation order O( 2 + h 2 ) :

(5.8)

Since the difference equation (5.8) can be written in the divergence nominal form

then, scalarly multiplying equation (5.9) by 1, we obtain

therefore, the following relation holds:

which can be considered a grid analog of the first conservation law.niya. Thus, scheme (5.8) is conservative. ATit is proved that the scheme (5.8) isL 2 -conservative and its decisionsatisfies the grid analogue of the integral conservation law

5.3. Implicit difference schemes (review).In this paragraph, weLet us consider implicit difference schemes for the Korteweg-de Vries equation.

Variant of the two-layer scheme - implicit absolutely stable schemema with order of approximation O ( 2 , h 4 ) :

The solution of the difference scheme (3.29) is calculated using seven diagonal cyclic sweep. The question of conservatismthis scheme has not been studied.

The paper proposes an implicit three-layer scheme with weights:

(5.10)

Difference scheme (5.10) with space-periodic solutions is conservative,L 2 - conservative at =1/2 and  =1/4 for her solutions, there are grid analogues of the integralconservation laws.

6. Numerical solution

The numerical solution for (3.2), (3.3), (3.4) was done using the explicit scheme

The initial-boundary value problem was solved on the segment . The function was taken as the initial conditions

u 0 (x)=sin (x).

The solution was obtained explicitly.

The calculation program was written in Turbo Pascal 7.0. The text of the main parts of the program is attached.

The calculations were carried out on a computer with an AMD -K 6-2 300 MHz processor with 3DNOW! technology, RAM size 32 MB.

7. Conclusion

This work is devoted to the study of the Korteweg – de Vries equation. An extensive literature review on the research topic was carried out. Various difference schemes for the KdV equation are studied. A practical calculation was performed using an explicit five-point spacing scheme

As the analysis of literature sources has shown, explicit schemes for solving KdV-type equations are most applicable. In this work, the solution was also obtained using an explicit scheme.

8. Literature

1. Landsberg G.S. Elementary textbook of physics. M.: Nauka, 1964. T. 3.

2. Feynman R., Layton R., Sands M. Feynman Lectures on Physics. M.: Mir, 1965. Issue 4.

3. Filippov A. G. Many-sided soliton. M.: Nauka, 1986. (B-chka "Quantum"; Issue 48).

4. Rubankov V.N. Solitons, new in life, science, technology. M.: Knowledge, 1983. (Physics; Issue 12).

5. Korteweg D.J., de Vries G. On the change form of long waves advancing in a rectangular channel and on new type of long stationary waves.//Phyl.May. 1895.e5. P. 422-443.

6. Sagdeev R.Z. Collective processes and shock waves in a rarefied plasma.-In the book: Questions of Plasma Theory, Issue 4. M.: Atomiz-dat, 1964, p.20-80.

7. Berezin Yu.A., Karpman V.I. On the theory of non-stationary waves of finite amplitude in a rarefied plasma. // ZhETF, 1964, v.46, issue 5, p. 1880-1890.

8. Zabusky N.J., Kruskal M.D. Interactions of "solitons" in a collisionless plasma and the reccurence of initial states // Phys.Rev.Lett. 1965.V.fifteen. fuck R.240-243.

9. Bullaf R., Caudrey F. Solitons. M.: Mir; 1983

10. Sjoberg A. On the Korteweg-de Vries equation, existence and uniqueness, Uppsala University, Department of Computers, 1967

11. Temam R. Sur un probleme non lineare // J. Math. Pures Anal. 1969, V.48, 2, P. 159-172.

12. Lyon J.-L. Some methods for solving nonlinear boundary value problems. M.: Mir, 1972.

13. Kruzhkov S.N. Faminsky A.V. Generalized solutions for the Korteweg-de Vries equation.// Matem. collection, 1983, vol. 120(162), e3, pp. 396-445

14. Gardner C.S., Green J.M., Kruskal M.D., Miura R.M. Method for solving the Korteweg-de Vries equation // Phys.Rev.Lett. 1967. V. 19.P. 1095-1097.

15. Shabat A.B. On the Korteweg-de Vries equation // DAN USSR, 1973, vol. 211, eb, pp. 1310-1313.

16. Faminsky A.V. Boundary value problems for the Korteweg-de Vries equation and its generalizations: Diss. Phys.-Math. Sciences, M: RUDN University, 2001

17. Miura R.M., Gardner C.S., Kruscal M.D. Korteweg-de Vries equation and generlization. II. Existence of conservation laws and constants of motion. // J.Math.Phys. 1968. V.9. P. 1204-1209.

18. Amosov A.A., Zlotnik A.A. Difference scheme for equations of gas motions.

19. Samarskii A.A., Mazhukin V.I., Matus P.P., Mikhailik I.A. Z /2-conservative schemes for the Korteweg-de Vries equation.// DAN, 1997, v.357, e4, pp.458-461

20. Berezin Yu.A. Modeling of nonlinear wave processes. Novosibirsk: Science. 1982.

21. Berezin Yu.A., On numerical solutions of the Korteweg-de Vries equation.// Numerical methods of continuum mechanics. Novosibirsk, 1973, v.4, e2, p.20-31

22. Samarskii A.A., Nikolaev Methods for solving grid equations. M: Science, 1978

23. Samarskii A.A., Gulin A.V. Numerical methods. M: Science, 1989

24. Bakhvalov N.S., Zhidkov N.P., Kobelkov G.M. Numerical methods. M: Science, 1987

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. AT last years of his life he completed the 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 of the medium; with dispersion, different sections of the wave move at different speeds 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) was published Enrico Fermi, John Pasta and Stan Ulam, dedicated to research 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 magnetic flux quantum, 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

Sailors have long known high-altitude solitary waves that destroy ships. For a long time it was believed that this occurs only in the open ocean. However, recent data suggest that solitary killer waves (up to 20-30 meters high), or solitons (from the English solitary - “solitary”), can also appear in coastal zones. The Birmingham Incident We were about 100 miles southwest of Durban on our way to Cape Town. The cruiser was moving quickly and with little to no rolling, encountering moderate swell and wind waves, when suddenly we fell into a hole and rushed down into the next wave, which swept through the first gun turrets and collapsed on our open captain's bridge.I was knocked down and, at a height of 10 meters above sea level, found myself in a half-meter layer of water.The ship experienced such a blow that many thought that we were torpedoed.The captain immediately reduced move, but this precaution was in vain, as moderate sailing conditions were restored and no more "pits" came across. This incident, which happened at night with a darkened ship, was one of the most exciting at sea. I readily believe that a loaded ship under such circumstances can drown". This is how a British officer from the cruiser "Birmingham-. This story took place during the Second World War, so the reaction of the crew, who decided that the cruiser was torpedoed, is understandable. A similar incident with the steamer Huarita in 1909 did not end so well. It carried 211 passengers and crew. All died. Such single waves unexpectedly appearing in the ocean, in fact, are called killer waves, or solitons. It would seem that. any storm can be called a killer .. Indeed, how many ships died during the storm and are dying now? How many sailors found their last resting place in the depths of the raging sea? And yet the waves. resulting from sea storms and even hurricanes are not called "killers". It is believed that an encounter with a soliton is most likely off the southern coast of Africa. When the shipping lanes changed due to the Suez Canal and ships stopped sailing around Africa, the number of encounters with killer waves decreased. Nevertheless, already after the Second World War, since 1947, in about 12 years, very large ships, the Bosfontein, met with solitons. "Giasterkerk", "Orinfontein" and "Jacherefontein", not counting the smaller local vessels. During the Arab-Israeli war, the Suez Canal was practically closed, and the movement of ships around Africa again became intense. From a meeting with a killer wave in June 1968, the World Glory supertanker with a displacement of more than 28 thousand tons died. The tanker received a storm warning, and when the storm approached, everything was carried out according to the instructions. Nothing bad was expected. But among the usual wind waves, which did not pose a serious danger. suddenly there was a huge wave about 20 meters high with a very steep front. She lifted the tanker so that its middle rested on the wave, and the bow and stern were in the air. The tanker was loaded with crude oil and broke in half under its own weight. These halves remained buoyant for some time, but after four hours the tanker sank to the bottom. True, most of the crew managed to be saved. In the 70s, the "attacks" of killer waves on ships continued. In August 1973, the Neptune Sapphire, sailing from Europe to Japan, 15 miles from Cape Hermis, with a wind of about 20 meters per second, experienced an unexpected blow from a solitary wave that had come from nowhere. The blow was so strong that the bow of the ship, about 60 meters long, broke off from the hull! The ship "Neptune Sapphire" had the most advanced design for those years. Nevertheless, the meeting with the killer wave turned out to be fatal for him. Quite a few such cases have been described. Naturally, not only large ships, on which there are possibilities for saving the crew, fall into the terrible list of disasters. Meeting with killer waves for small craft often ends much more tragically. Such ships not only experience the strongest blow. capable of destroying them, but on a steep leading edge, the waves can easily overturn. This is happening so fast that it is impossible to count on salvation. This is not a tsunami. What are these killer waves? The first thought that comes to the mind of an informed reader is a tsunami. After the catastrophic "raid" of gravitational waves on the southeastern coast of Asia, many imagine the tsunami as an eerie wall of water with a steep front, falling on the shore and washing away houses and people. Indeed, tsunamis are capable of much. After the appearance of this wave near the northern Kuriles, hydrographers, studying the consequences, discovered a decent-sized boat thrown over the coastal hills into the interior of the island. That is, the energy of the tsunami is simply amazing. However, this is all about tsunamis that “attack” the coast. Translated into Russian, the term "tsunami" means "big wave in the harbor." It is very difficult to find it in the open ocean. There, the height of this wave usually does not exceed one meter, and the average, typical dimensions are tens of centimeters. And the slope is extremely small, because at such a height its length is several kilometers. So it is almost impossible to detect a tsunami against the background of running wind waves or swell. Why, then, when “attacking” a shore, tsunamis become so frightening? The fact is that this wave, due to its large length, sets the water in motion throughout the entire depth of the ocean. And when it reaches relatively shallow areas during its spreading, all this colossal mass of water rises from the depths. This is how a “harmless” wave in the open ocean becomes destructive on the coast. So killer waves are not tsunamis. In fact, solitons are an unusual and little-studied phenomenon. They are called waves, although in fact they are something else. For the emergence of solitons, of course, some initial impulse, an impact, is needed, otherwise where will the energy come from, but not only. Unlike conventional waves, solitons propagate over long distances with very little energy dissipation. This is a mystery that is yet to be explored. Solitons practically do not interact with each other. As a rule, they propagate at different speeds. Of course, it may happen that one soliton catches up with the other, and then they are summed up in height, but then they still scatter along their paths again. Of course, the addition of solitons is a rare event. But there is another reason for the sharp increase in their steepness and height. This reason is the underwater ledges through which the soliton "runs". At the same time, energy is reflected in the underwater part, and the wave, as it were, “splashes” upwards. A similar situation was studied on physical models by an international scientific group. Based on these studies, safer ship routes can be laid. But there are still many more mysteries than studied features, and the mystery of killer waves is still waiting for its researchers. Particularly mysterious are the solitons inside the waters of the sea, on the so-called "density jump layer". These solitons can lead (or have already led) to submarine disasters.

After calculations and searching for analogies, these scientists found that the equation used by Fermi, Pasta and Ulam, with a decrease in the distance between the weights and with an unlimited increase in their number, goes into the Korteweg-de Vries equation. That is, in essence, the problem proposed by Fermi was reduced to the numerical solution of the Korteweg-de Vries equation, proposed in 1895 to describe a solitary Russell wave. Approximately in the same years, it was shown that the Korteweg-de Vries equation is also used to describe ion-acoustic waves in plasma. Then it became clear that this equation is found in many areas of physics and, therefore, the solitary wave, which is described by this equation, is a widespread phenomenon.

Continuing the computational experiments to model the propagation of such waves, Kruskal and Zabusky considered their collision. Let us dwell in more detail on the discussion of this remarkable fact. Let there be two solitary waves described by the Korteweg-de Vries equation, which differ in amplitude and move one after the other in the same direction (Fig. 2). It follows from the formula for solitary waves (8) that the higher the velocity of such waves, the greater their amplitude, and the peak width decreases with increasing amplitude. Thus, high solitary waves move faster. A wave with a larger amplitude will overtake a wave with a smaller amplitude moving ahead. Then, for some time, the two waves will move together as a whole, interacting with each other, and then they will separate. A remarkable property of these waves is that after their interaction, the form and

Rice. 2. Two solitons described by the Korteweg-de Vries equation,

before interaction (top) and after (bottom)

the speed of these waves is restored. Both waves after the collision are only displaced by a certain distance compared to how they would move without interaction.

The process, in which the shape and speed are preserved after the interaction of waves, resembles an elastic collision of two particles. Therefore, Kruskal and Zabuski called such solitary waves solitons (from the English solitary - solitary). This special name for solitary waves, consonant with the electron, proton and many other elementary particles, is currently generally accepted.

Solitary waves, which were discovered by Russell, indeed behave like particles. A large wave does not pass through a small one during their interaction. When solitary waves touch, the large wave slows down and decreases, and the wave that was small, on the contrary, accelerates and grows. And when the small wave grows to the size of a large one, and the large one decreases to the size of a small one, the solitons separate and the larger one moves forward. Thus, solitons behave like elastic tennis balls.

Let's give a definition of a soliton. Soliton called a non-linear solitary wave, which retains its shape and speed during its own movement and collision with similar solitary waves, that is, it is a stable formation. The only result of the interaction of solitons can be some phase shift.

The discoveries related to the Korteweg-de Vries equation did not end with the discovery of the soliton. The next important step related to this remarkable equation was the creation of a new method for solving non-linear partial differential equations. It is well known that finding solutions to nonlinear equations is very difficult. Until the 1960s, it was believed that such equations could only have certain particular solutions that satisfy specially given initial conditions. However, the Korteweg-de Vries equation also found itself in an exceptional position in this case.

In 1967, American physicists K.S. Gardner, J.M. Green, M. Kruskal and R. Miura have shown that the solution of the Korteweg-de Vries equation can in principle be obtained for all initial conditions that vanish in a certain way as the coordinate tends to infinity. They used the transformation of the Korteweg-de Vries equation to a system of two equations, now called the Lax pair (after the American mathematician Peter Lax, who made a great contribution to the development of the theory of solitons), and discovered a new method for solving a number of very important non-linear partial differential equations. This method is called the method of the inverse scattering problem, since it essentially uses the solution of the problem of quantum mechanics about the reconstruction of the potential from scattering data.

2.2. Group soliton

Above, we said that in practice the waves, as a rule, propagate in groups. Similar groups of waves on the water people have observed since time immemorial. T. Benjamin and J. Feyer managed to answer the question of why "flocks" of waves are so typical for waves on water, only in 1967. By theoretical calculations, they showed that a simple periodic wave in deep water is unstable (now this phenomenon is called the Benjamin-Fejér instability), and therefore the waves on the water are divided into groups due to instability. The equation that describes the propagation of wave groups on water was obtained by V.E. Zakharov in 1968. By that time, this equation was already known in physics and was called the nonlinear Schrödinger equation. In 1971, V.E. Zakharov and A.B. Shabat showed that this nonlinear equation also has solutions in the form of solitons, moreover, the nonlinear Schrödinger equation, as well as the Korteweg-de Vries equation, can be integrated by the method of the inverse scattering problem. The solitons of the nonlinear Schrödinger equation differ from the Korteweg-de Vries solitons discussed above in that they correspond to the shape of the wave group envelope. Outwardly, they resemble modulated radio waves. These solitons are called group solitons and sometimes envelope solitons. This name reflects the persistence in the interaction of the wave packet envelope (analogous to the dashed line shown in Fig. 3), although the waves themselves under the envelope move at a speed different from the group speed. In this case, the shape of the envelope is described

Rice. 3. An example of a group soliton (dashed line)

addiction

a(x,t)=a 0 ch -1 ()

where a a - amplitude, and l is half the size of the soliton. Usually, there are from 14 to 20 waves under the envelope of a soliton, with the middle wave being the largest. Related to this is the well-known fact that the highest wave in a group on water is between the seventh and tenth (the ninth wave). If a larger number of waves has formed in a group of waves, then it will break up into several groups.

The nonlinear Schrödinger equation, like the Korteweg-de Vries equation, is also widely used in the description of waves in various fields of physics. This equation was proposed in 1926 by the outstanding Austrian physicist E. Schrödinger to analyze the fundamental properties of quantum systems and was originally used to describe the interaction of intraatomic particles. The generalized or nonlinear Schrödinger equation describes a set of phenomena in the physics of wave processes. For example, it is used to describe the effect of self-focusing when a powerful laser beam acts on a nonlinear dielectric medium and to describe the propagation of nonlinear waves in a plasma.

3. Statement of the problem

3.1. Description of the model. At present, there is a significantly growing interest in the study of nonlinear wave processes in various fields of physics (for example, in optics, plasma physics, radiophysics, hydrodynamics, etc.). To study waves of small but finite amplitude in dispersive media, the Korteweg-de Vries (KdV) equation is often used as a model equation:

u t + ii x + b and xxx = 0 (3.1)

The KdV equation was used to describe magnetosonic waves propagating strictly across the magnetic field or at angles close to

The main assumptions that are made when deriving the equation are: 1) small but finite amplitude, 2) the wavelength is large compared to the dispersion length.

Compensating the effect of nonlinearity, dispersion makes it possible to form in a dispersive medium stationary waves of finite amplitude - solitary and periodic. The solitary waves for the KdV equation came to be called solitons after the work. Periodic waves are called cnoidal waves. The corresponding formulas for their description are given in.

3.2. Formulation of a differential problem. In this paper, we study the numerical solution of the Cauchy problem for the Korteweg-de Vries equation with periodic conditions in space in a rectangle Q T ={( t , x ):0< t < T , x Î [0, l ].

u t + ii x + b and xxx = 0 (3.2)

u(x,t)| x=0 =u(x,t)| x=l (3.3)

with initial condition

u(x,t)| t=0 =u 0 (x) (3.4)

4. Properties of the Korteweg - de Vries equation

4.1. A brief review of the results on the KdV equation. The Cauchy problem for the KdV equation under various assumptions about u 0 (X) considered in many works. The problem of the existence and uniqueness of a solution with periodicity conditions as boundary conditions was solved in this work using the finite difference method. Later, under less strong assumptions, the existence and uniqueness were proved in the article in the space L ¥ (0,T,H s (R ​​1)), where s>3/2, and in the case of a periodic problem, in the space L ¥ (0 ,T,H ¥ (C)) where C is a circle of length equal to the period, in Russian these results are presented in the book.