arXiv subjects

A. M. Kamchatnov

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

At least 37 records · Page 2Linked to original sources

arXiv2023

Modulation theory for the sine-Gordon equation

We give the solution of the Whitham modulation equations for envelopes of pulses evolving according to the sine-Gordon equation. The Whitham equations are interpreted as the equations of relativistic hydrodynamics and their solving is reduced by the hodograph method to solving a linear partial differential equation. We describe the class of solutions of this equation with separation of variables and illustrate the theory by an example of the nonlinear wave packet evolution accompanied by its shrinking and decrease of the number of oscillations in the Whitham nonlinear region.

nlin.PS↗

arXiv2022

Motion of dark solitons in a non-uniform flow of Bose-Einstein condensate

We study motion of dark solitons in a non-uniform one-dimensional flow of Bose-Einstein condensate. Our approach is based on Hamiltonian mechanics applied to the particle-like behavior of dark solitons in a slightly non-uniform and slowly changing surrounding. In one-dimensional geometry, the condensate's wave function undergoes the jump-like behavior across the soliton and this leads to generation of the counterflow in the background condensate. For correct description of soliton's dynamics, the contributions of this counterflow to the momentum and energy of the soliton are taken into account. The resulting Hamilton equations are reduced to the Newton-like equation for the soliton's path and this Newton equation is solved in several typical situations. The analytical results are confirmed by numerical calculations.

nlin.PS↗

arXiv2022

Contour dynamics of two-dimensional dark solitons

Equations for contour dynamics of dark solitons are obtained for the general form of the nonlinearity function. Their self-similar solution which describes the nonlinear stage of the bending instability of dark solitons is studied in detail.

nlin.PS↗

arXiv2021

Number of solitons produced from a large initial pulse in the generalized NLS dispersive hydrodynamics theory

We show that the number of solitons produced from an arbitrary initial pulse of the simple wave type can be calculated analytically if its evolution is governed by a generalized nonlinear Schrödinger equation provided this number is large enough. The final result generalizes the asymptotic formula derived for completely integrable nonlinear wave equations like the standard NLS equation with the use of the inverse scattering transform method.

nlin.PS↗

arXiv2021

Propagation of instability fronts in modulationally unstable systems

We study evolution of pulses propagating through focusing nonlinear media. Small disturbance on the smooth initial non-uniform background leads to formation of the region of strong nonlinear oscillations. We develop here an asymptotic method for finding the law of motion of the front of this region. The method is applied to the focusing nonlinear Schroedinger equation for the particular cases of Talanov and Akhmanov-Sukhorukov-Khokhlov initial distributions with zero initial phase. The approximate analytical results agree well with the exact numerical solutions for these two problems.

nlin.PS↗

arXiv2021

Quantitative Analysis of Shock Wave Dynamics in a Fluid of Light

We report on the formation of a dispersive shock wave in a nonlinear optical medium. We monitor the evolution of the shock by tuning the incoming beam power. The experimental observations for the position and intensity of the solitonic edge of the shock, as well as the location of the nonlinear oscillations are well described by recent developments of Whitham modulation theory. Our work constitutes a detailed and accurate benchmark for this approach. It opens exciting possibilities to engineer specific configurations of optical shock wave for studying wave-mean flow interaction.

cond-mat.quant-gas↗

arXiv2021

Gurevich-Pitaevskii problem and its development

We present an introduction to the theory of dispersive shock waves in the framework of the approach proposed by Gurevich and Pitaevskii (Zh. Eksp. Teor. Fiz., 65, 590 (1973) [Sov. Phys. JETP, 38, 291 (1974)]) based on the Whitham theory of modulation of nonlinear waves. We explain how Whitham equations for a periodic solution can be derived for the Korteweg-de Vries equation and outline some elementary methods to solve them. We illustrate this approach with solutions to the main problems discussed by Gurevich and Pitaevskii. We consider a generalization of the theory to systems with weak dissipation and discuss the theory of dispersive shock waves for the Gross-Pitaevskii equation.

nlin.PS↗

arXiv2021

Propagation of wave packets along intensive simple waves

We consider propagation of high-frequency wave packets along a smooth evolving background flow whose evolution is described by a simple-wave type of solutions of hydrodynamic equations. In geometrical optics approximation, the motion of the wave packet obeys the Hamilton equations with the dispersion law playing the role of the Hamiltonian. This Hamiltonian depends also on the amplitude of the background flow obeying the Hopf-like equation for the simple wave. The combined system of Hamilton and Hopf equations can be reduced to a single ordinary differential equation whose solution determines the value of the background amplitude at the location of the wave packet. This approach extends the results obtained in Ref.~\cite{ceh-19} for the rarefaction background flow to arbitrary simple-wave type background flows. The theory is illustrated by its application to waves obeying the KdV equation.

nlin.PS↗

arXiv2020

Theory of quasi-simple dispersive shock waves and number of solitons evolved from a nonlinear pulse

The theory of motion of edges of dispersive shock waves generated after wave breaking of simple waves is developed. It is shown that this motion obeys Hamiltonian mechanics complemented by a Hopf-like equation for evolution of the background flow that interacts with edge wave packets or edge solitons. A conjecture about existence of a certain symmetry between equations for the small-amplitude and soliton edges is formulated. In case of localized simple wave pulses propagating through a quiescent medium this theory provided a new approach to derivation of an asymptotic formula for the number of solitons produced eventually from such a pulse.

nlin.PS↗

arXiv2020

Plasma slab expansion into vacuum

The problem of collisionless plasma slab expansion into vacuum is solved within a two-temperature hydrodynamic approximation in the dispersionless limit of zero Debye radius. In the framework of such an approach, the solution by the Riemann method provides quite accurate description of the whole process of plasma dynamics. It is shown that the dispersionless approximation agrees very well with exact numerical solution of the full system of plasma hydrodynamic equations.

nlin.PS↗

arXiv2020

Dispersionless evolution of inviscid nonlinear pulses

We consider the one-dimensional dynamics of nonlinear non-dispersive waves. The problem can be mapped onto a linear one by means of the hodograph transform. We propose an approximate scheme for solving the corresponding Euler-Poisson equation which is valid for any kind of nonlinearity. The approach is exact for monoatomic classical gas and agrees very well with exact results and numerical simulations for other systems. We also provide a simple and accurate determination of the wave breaking time for typical initial conditions.

nlin.PS↗

arXiv2019

Wave breaking and formation of dispersive shock waves in a defocusing nonlinear optical material

We theoretically describe the quasi one-dimensional transverse spreading of a light pulse propagating in a nonlinear optical material in the presence of a uniform background light intensity. For short propagation distances the pulse can be described within a nondispersive approximation by means of Riemann's approach. For larger distances, wave breaking occurs, leading to the formation of dispersive shocks at both ends of the pulse. We describe this phenomenon within Whitham modulation theory, which yields an excellent agreement with numerical simulations. Our analytic approach makes it possible to extract the leading asymptotic behavior of the parameters of the shock.

nlin.PS↗

arXiv2019

Landau-Khalatnikov problem in relativistic hydrodynamics

An alternative approach to solving the Landau-Khalatnikov problem on one-dimensional stage of expansion of hot hadronic matter created in collisions of high-energy particles or nuclei is suggested. Solving the relativistic hydrodynamics equations by the Riemann method yields a representation for Khalatnikov's potential which satisfies explicitly the condition of symmetry of the matter flow with respect to reflection in the central plane of the initial distribution of matter. New exact relationships are obtained for evolution of the density of energy in the center of the distribution and for laws of motion of boundaries between the general solution and the rarefaction waves. The rapidity distributions are derived in the Landau approximation with account of the pre-exponential factor.

nlin.PS↗

arXiv2019

Evolution of intensive light pulses in a nonlinear medium with the Raman effect

In this paper, we study the evolution of intensive light pulses in nonlinear single-mode fibers. The dynamics of light in such fibers is described by the nonlinear Schrödinger equation with the Raman term, due to stimulated Raman self-scattering. It is shown that dispersive shock waves are formed during the evolution of sufficiently intensive pulses. In this case, the situation is much richer than for the nonlinear Schrödinger equation with Kerr nonlinearity only. The Whitham equations are obtained under the assumption that the Raman term can be considered as a small perturbation. These equations describe slow evolution of dispersive shock waves. It is shown that if one takes into account the Raman effect, then dispersive shock waves can asymptotically acquire a stationary profile. The analytical theory is confirmed by numerical calculations.

nlin.PS↗

arXiv2019

Evolution of wave pulses in fully nonlinear shallow-water theory

We consider evolution of wave pulses with formation of dispersive shock waves in framework of fully nonlinear shallow-water equations. Situations of initial elevations or initial dips on the water surface are treated and motion of the dispersive shock edges is studied within the Whitham theory of modulations. Simple analytical formulas are obtained for asymptotic stage of evolution of initially localized pulses. Analytical results are confirmed by exact numerical solutions of the fully nonlinear shallow-water equations.

nlin.PS↗

arXiv2019

Short-distance propagation of nonlinear optical pulses

We theoretically describe the quasi one-dimensional transverse spreading of a light pulse propagating in a defocusing nonlinear optical material in the presence of a uniform background light intensity. For short propagation distances the pulse can be described within a nondispersive approximation by means of Riemann's approach. The theoretical results are in excellent agreement with numerical simulations.

nlin.PS↗

arXiv2018

Dispersive shock waves theory for non-integrable equations

We suggest a method for calculation of parameters of dispersive shock waves in framework of Whitham modulation theory applied to non-integrable wave equations with a wide class of initial conditions corresponding to propagation of a pulse into a medium at rest. The method is based on universal applicability of Whitham's `number of waves conservation law' as well as on the conjecture of applicability of its soliton counterpart to the above mentioned class of initial conditions which is substantiated by comparison with similar situations in the case of completely integrable wave equations. This allows one to calculate the limiting characteristic velocities of the Whitham modulation equations at the boundary with the smooth part of the pulse whose evolution obeys the dispersionless approximation equations. We show that explicit analytic expressions can be obtained for laws of motion of the edges. The validity of the method is confirmed by its application to similar situations described by the integrable Korteweg-de Vries (KdV) and nonlinear Schrödinger (NLS) equations and by comparison with the results of numerical simulations for the generalized KdV and NLS equations.

nlin.PS↗

arXiv2018

Wave Breaking in Dispersive Fluid Dynamics of the Bose-Einstein Condensate

The problem of wave breaking during its propagation in the Bose-Einstein condensate to a stationary medium is considered for the case when the initial profile at the breaking instant can be approximated by a power function of the form $(-x)^{1/n}$. The evolution of the wave is described by the Gross-Pitaevskii equation so that a dispersive shock wave is formed as a result of breaking; this wave can be represented using the Gurevich-Pitaevskii approach as a modulated periodic solution to the Gross-Pitaevskii equation, and the evolution of the modulation parameters is described by the Whitham equations obtained by averaging the conservation laws over fast oscillations in the wave. The solution to the Whitham modulation equations is obtained in closed form for $n = 2,3$, and the velocities of the dispersion shock wave edges for asymptotically long evolution times are determined for arbitrary integer values $n > 1$. The problem considered here can be applied for describing the generation of dispersion shock waves observed in experiments with the Bose-Einstein condensate.

nlin.PS↗