arXiv subjects

A. M. Kamchatnov

Publications and source records attributed to A. M. Kamchatnov.

At least 19 recordsLinked to original sources

arXiv2026

Evolution of instability fronts in sine-Gordon equation dynamics

Solutions of the Whitham modulation equations for one-phase periodic waves obeying the sine-Gordon equation are found that describe the evolution of an oscillatory region behind an instability front propagating into the instability region. A simple self-similar solution describes the whole region between two fronts of instability resulting from a localized initial disturbance in the unstable state. Another hodograph solution represents typical waves close to the instability fronts. This theory generalizes the approach used previously for systems obeying the nonlinear Schroedinger equation.

nlin.PS↗

arXiv2026

Evolution of localized pulses in the defocusing modified Korteweg-de Vries equation theory

In this work, we develop, in the Gurevich-Pitaevskii framework, an analytic theory for the evolution of localized pulses in the defocusing modified Korteweg-de Vries equation theory for situations when a dispersive shock does not eventually transform into a sequence of well-separated solitons. We found solutions to the Whitham modulation equations for the corresponding so-called "quasi-simple" dispersive shock waves and illustrated this solution with concrete examples of an initial pulse. Comparison of the analytical solution with direct numerical simulations showed that the modulation theory provides a very accurate description of the wave pattern even at one wavelength scale.

nlin.PS↗

arXiv2026

`Relativistic' propagation of instability fronts in nonlinear Klein-Gordon equation dynamics

We consider propagation of instability fronts in conservative nonlinear wave systems by the Whitham method. Whitham modulation equations for periodic solutions of the generalized Klein-Gordon equation are solved in the limit of asymptotically large times, when the size of the instability wave region is much greater than the size of the initial localized disturbance, so the solution reaches the self-similar regime. The general self-similar solution is illustrated by two typical examples of the nonlinearity function. It is shown that in these models the instability fronts propagate with maximal group velocity.

nlin.PS↗

arXiv2025

Supersonic Flow Past an Obstacle in a Quasi-Two-Dimensional Lee-Huang-Yang Quantum Fluid

A supersonic flow past an obstacle can generate a rich variety of wave excitations. This paper investigates, both analytically and numerically, two types of excitations generated by the flow of a Lee-Huang-Yang quantum fluid past an obstacle: linear radiation and oblique dark solitons. We show that wave crests of linear radiation can be accurately described by the proper modification of the Kelvin original theory, while the oblique dark soliton solution is obtained analytically by transformation of the 1D soliton solution to the obstacle's reference frame. A comparison between analytical predictions and numerical simulations demonstrates good agreement, validating our theoretical approach.

cond-mat.quant-gas↗

arXiv2025

Theory of dispersive shock waves induced by the Raman effect in optical fibers

We develop the theory of dispersive shock waves in optical fibers for the case of long-distance propagation of optical pulses, when the small Raman effect stabilizes the profile of the shock. The Whitham modulation equations are derived as the basis for the Gurevich-Pitaevskii approach to the analytical theory of such shocks. We show that the wave variables at both sides of the shock are related by the analogue of the Rankine-Hugoniot condition that follows from the conservation laws of the Whitham equations. Solutions of the Whitham equations yield the profiles of the wave variables that agree very well with the exact numerical solution of the generalized nonlinear Schroedinger equation for propagation of optical pulses.

nlin.PS↗

arXiv2025

Dynamics of ring solitons in an expanding cloud of a Bose-Einstein condensate

In this paper, we derive equations for the dynamics of ring dark solitons in an expanding cloud of a two-dimensional Bose-Einstein condensate. Assuming that the soliton's width is much smaller than its radius, we obtain the Hamilton equations for its evolution. Then they are transformed into the Newton equation, which is more convenient for applications. The general theory is illustrated by the solution of the Newton equation for the case of the axially symmetric condensate cloud, which expands after switching off a harmonic trap. The validity of our approximate analytical approach is confirmed by comparison with the results of numerical simulations of the Gross-Pitaevskii equation.

nlin.PS↗

arXiv2025

Asymptotic integrability and its consequences

We give a brief review of the concept of asymptotic integrability, which means that the Hamilton equations for the propagation of short-wavelength packets along a smooth, large-scale background wave have an integral independent of the initial conditions. The existence of such an integral leads to a number of important consequences, which include, besides the direct application to the packets propagation problems, Hamiltonian theory of narrow solitons motion and generalized Bohr-Sommerfeld rule for parameters of solitons produced from an intensive initial pulse. We show that in the case of systems with two wave variables and exact fulfillment of the asymptotic integrability condition, the `quantization' of mechanical systems, associated with the additional integrals, yields the Lax pairs for a number of typical completely integrable equations, and this sheds new light on the origin of the complete integrability in nonlinear wave physics.

nlin.SI↗

arXiv2024

Asymptotic integrability and Hamilton theory of soliton's motion along large-scale background waves

We consider the problem of soliton-mean field interaction for the class of asymptotically integrable equations, where the notion of the asymptotic integrability means that the Hamilton equations for the high-frequency wave packet's propagation along a large-scale background wave have an integral of motion. Using the Stokes remark, we transform this integral to the integral for the soliton's equations of motion and then derive the Hamilton equations for the soliton's dynamics in a universal form expressed in terms of the Riemann invariants for the hydrodynamic background wave. The physical properties are specified by the concrete expressions for the Riemann invariants. The theory is illustrated by its application to the soliton's dynamics which is described by the Kaup-Boussinesq system.

nlin.SI↗

arXiv2024

Asymptotic integrability of nonlinear wave equations

We introduce the notion of asymptotic integrability into the theory of nonlinear wave equations. It means that the Hamiltonian structure of equations describing propagation of high-frequency wave packets is preserved by hydrodynamic evolution of the large-scale background wave, so that these equations have an additional integral of motion. This condition is expressed mathematically as a system of equations for the carrier wave number as a function of the background variables. We show that a solution of this system for a given dispersion relation of linear waves is related with the quasiclassical limit of the Lax pair for the completely integrable equation having the corresponding dispersionless and linear dispersive behavior. We illustrate the theory by several examples.

nlin.SI↗

arXiv2024

Hamiltonian mechanics of "magnetic'' solitons in two-component Bose-Einstein condensates

We consider motion of a "magnetic'' soliton in two-component condensates along a non-uniform and time-dependent backgrounds in framework of the Hamiltonian mechanics. Our approach is based on generalization of Stokes' remark that soliton's velocity is related with its inverse half-width by the dispersion law for linear waves continued to the region of complex wave numbers. We obtain expressions for the canonical momentum and the Hamiltonian as functions of soliton's velocity and transform the Hamilton equations to the Newton-like equation. The theory is illustrated by several examples of concrete soliton's dynamics.

nlin.PS↗

arXiv2024

Propagation of dark solitons of DNLS equation along a large-scale background

We study dynamics of dark solitons in the theory of the DNLS equation by the method based on imposing the condition that this dynamics must be Hamiltonian. Combining this condition with Stokes' remark that relationships for harmonic linear waves and small-amplitude soliton tails satisfy the same linearized equations, so the corresponding solutions can be converted one into the other by replacement of the packet's wave number $k$ by $iκ$, $κ$ being the soliton's inverse half-width, we find the Hamiltonian and the canonical momentum of the soliton's motion. The Hamilton equations are reduced to the Newton equation whose solutions for some typical situations are compared with exact numerical solutions of the DNLS equation.

nlin.PS↗

arXiv2024

Hamilton theory of NLS equation soliton motion

We suggest the method of derivation of Hamilton equations which describe the motion of solitons along non-uniform and time dependent large-scale background in case of wave dynamics described by the completely integrable equations in the Ablowitz-Kaup-Newell-Segur scheme. The method is based on development of old Stokes' argumentation which allows one to continue analytically some relationships derived for linear waves to the soliton region. It is presented here for a particular case of the defocusing nonlinear Schrödinger equation. We formulate the condition when the external potential should only be taken into account for the background evolution, and in this case we obtain the Newton equation for the soliton dynamics.

nlin.PS↗

arXiv2024

Propagation of generalized Korteweg-de Vries solitons along large-scale waves

We consider propagation of solitons along large scale background waves in the generalized Korteweg-de Vries (gKdV) equation theory when the width of the soliton is mach smaller than the characteristic size of the background wave. Due to this difference in scales, the soliton's motion does not affect the dispersionless evolution of the background wave. We obtained the Hamilton equations for soliton's motion and derived simple relationships which express the soliton's velocity in terms of a local value of the background wave. Solitons' paths obtained by integration of these relationships agree very well with the exact numerical solutions of the gKdV equation.

nlin.PS↗

arXiv2023

Quasiclassical integrability condition in AKNS scheme

In this paper, we study the condition of quasiclassical integrability of soliton equations. This condition states that the Hamiltonian structure of equations, which govern propagation of high-frequency wave packets, is preserved by the dispersionless flow independently of initial conditions. If this condition is fulfilled, then the carrier wave number of any packet is a certain function of local values of the dispersionless variables pertained to the soliton equations under consideration. We show for several examples that this function together with the dispersion relation for linear harmonic waves determine the quasiclassical limit of the Lax pair functions in the scalar representation of the Ablowitz-Kaup-Newell-Segur scheme.

nlin.SI↗

arXiv2023

Undular bore theory for the modified Korteweg-de Vries-Burgers equation

We consider nonlinear wave structures described by the modified Korteweg-de Vries equation with taking into account a small Burgers viscosity for the case of step-like initial conditions. The Whitham modulation equations are derived which include the small viscosity as a perturbation. It is shown that for long enough time of evolution this small perturbation leads to stabilization of cnoidal bores and their main characteristics are obtained. Applicability conditions of this approach are discussed. Analytical theory is compared with numerical solutions and good agreement is found.

nlin.PS↗

arXiv2023

Propagation of wave packets along large-scale background waves

We study propagation of high-frequency wave packets along a large-scale background wave which evolves according to dispersionless hydrodynamic equations for two variables (fluid density and flow velocity). Influence of the wave packet on evolution of the background wave is neglected, so the large-scale evolution can be found independently of the wave packet's motion. At the same time, propagation of the packet depends in essential way on the background wave and it can be considered in framework of geometric optics approximation with the use of Hamilton equations for the carrier wave number and the mean coordinate of the packet. We derive equations for the carrier wave number as a function of the parameters which describe the background wave. When they are solved, the path of the packet can be found by simple integration of the Hamilton equation. The theory is illustrated by its application to the problem of propagation of wave packets along expanding large-scale wave which evolution is described by the shallow water equations. In particular, they correspond to the dispersionless limit of the defocusing nonlinear Schrödinger equation, and then the expanding wave can be considered as an expanding cloud of the Bose-Einstein condensate. Reflection of wave packets from upstream flows and their propagation along stationary flows are also discussed. The analytical solutions found for these particular cases agree very well with exact numerical solution of the nonlinear Schrödinger equation.

nlin.PS↗

arXiv2023

Asymptotic theory of not completely integrable soliton equations

We develop the theory of transformation of intensive initial nonlinear wave pulses to trains of solitons emerging at asymptotically large time of evolution. Our approach is based on the theory of dispersive shock waves in which the number of nonlinear oscillations in the shock becomes the number of solitons at the asymptotic state. We show that this number of oscillations, which is proportional to the classical action of particles associated with the small-amplitude edges of shocks, is preserved by the dispersionless flow. Then the Poincaré-Cartan integral invariant is also constant and therefore it reduces to the quantization rule similar to the Bohr-Sommerfeld quantization rule for linear spectral problem associated with completely integrable equations. This rule yields a set of `eigenvalues' which are related with the asymptotic solitons' velocities and other their characteristics. Our analytical results agree very well with the results of numerical solutions of the generalized nonlinear Schrödinger equation.

nlin.PS↗

arXiv2023

Dynamics of Interaction of Two Soliton Clouds

On the basis of relationship between the kinetic equation for two soliton clouds in the theory of the Korteweg-de Vries equation and equations of the Chaplygin gas dynamics it is shown that the existence of waves propagating without a change in their form is a fundamental property of the nonlinear dynamics of soliton gases. The solutions of several typical problems in the soliton gas dynamics are considered and characteristic features of such dynamics, which make it possible to estimate the effects of interaction of soliton gases, are indicated.

nlin.PS↗