arXiv subjects

Vladimir Kotlyarov

Publications and source records attributed to Vladimir Kotlyarov.

4 recordsLinked to original sources

arXiv2019

Dispersive Shock Wave, Generalized Laguerre Polynomials and Asymptotic Solitons of the Focusing Nonlinear Schrödinger Equation

We consider dispersive shock wave to the focusing nonlinear Schrödinger equation generated by a discontinuous initial condition which is periodic or quasi-periodic on the left semi-axis and zero on the right semi-axis. As an initial function we use a finite-gap potential of the Dirac operator given in an explicit form through hyper-elliptic theta-functions. The paper aim is to study the long-time asymptotics of the solution of this problem in a vicinity of the leading edge, where a train of asymptotic solitons are generated. Such a problem was studied in \cite{KK86} and \cite{K91} using Marchenko's inverse scattering technics. We investigate this problem exceptionally using the Riemann-Hilbert problems technics that allow us to obtain explicit formulas for the asymptotic solitons themselves that in contrast with the cited papers where asymptotic formulas are obtained only for the square of absolute value of solution. Using transformations of the main RH problems we arrive to a model problem corresponding to the parametrix at the end points of continuous spectrum of the Zakharov-Shabat spectral problem. The parametrix problem is effectively solved in terms of the generalized Laguerre polynomials which are naturally appeared after appropriate scaling of the Riemann-Hilbert problem in a small neighborhoods of the end points of continuous spectrum. Further asymptotic analysis give an explicit formula for solitons at the edge of dispersive wave. Thus, we give the complete description of the train of asymptotic solitons: not only bearing envelope of each asymptotic soliton, but its oscillating structure are found explicitly. Besides the second term of asymptotics describing an interaction between these solitons and oscillating background is also found. This gives the fine structure of the edge of dispersive shock wave.

math-ph↗

arXiv2015

Modified Korteweg-de Vries equation: modulated elliptic wave and a train of asymptotic solitons

We study the long-time asymptotic behavior of the Cauchy problem for the modified Korteweg - de Vries equation with an initial function of the step type. This function rapidly tends to zero as $x\to+\infty$ and to some positive constant c as $x\to-\infty$. In 1989 E. Khruslov and V. Kotlyarov have found that for a large time the solution breaks up into a train of asymptotic solitons located in the domain $4c^2t-C_N \ln t<x\leq4c^2t$ ($C_N$ is a constant). The number N of these solitons grows unboundedly as $t\to\infty$. In 2010 V. Kotlyarov and A. Minakov have studied temporary asymptotics of the solution of the Cauchy problem on the whole line and have found that in the domain $-6c^2 t<x<4c^2t$ this solution is described by a modulated elliptic wave. We considere here the modulated elliptic wave in the domain $4c^2t-C_N \ln t<x<4c^2t$. Our main result shows that the modulated elliptic wave also breaks up into solitons, which are similar to the asymptotic solitons, but differ from them in phase. It means that the modulated elliptic wave does not represent the asymptotics of the solution in the domain $4c^2t-C_N \ln t<x<4c^2t$. The correct asymptotic behavior of the solution is given by the train of asymptotic solitons. However, in the asymptotic regime as $t\to\infty$ in the region $4c^2t-\frac{N+1/4}{c}\ln t<x<4c^2t-\frac{N-3/4}{c}\ln t$ we can watch precisely a pair of solitons with numbers N. One of them is the asymptotic soliton while the other soliton is generated from the elliptic wave. Their phases become closer to each other for a large N, i.e. these solitons are also close to each other. This result gives the answer on a very important question about matching of the asymptotic formulas in the mentioned region where the both formulas are well-defined. We have a new and earlier unknown mechanism of matching of the asymptotics of the solution in the adjacent regions.

math-ph↗

arXiv2014

Periodic problem for the nonlinear Schroedinger equation

The paper offers the method of discovering of some class of solutions for the nonlinear Schroedinger equation. An algorithm of constructive solving of the Cauchy periodic problem with a finite-gap initial condition was also obtained.

nlin.SI↗

arXiv2003

Characteristic properties of the scattering data for the mKdV equation on the half-line

In this paper we describe characteristic properties of the scattering data of the compatible eigenvalue problem for the pair of differential equations related to the modified Korteweg-de Vries (mKdV) equation whose solution is defined in some half-strip $(0<x<\infty)\times[0,T]$, or in the quarter plane $(0<x<\infty)\times(0<t<\infty)$. We suppose that this solution has a $C^{\infty}$ initial function vanishing as $x\to\infty$, and $C^{\infty}$ boundary values, vanishing as $t\to\infty$ when $T=\infty$. We study the corresponding scattering problem for the compatible Zakharov-Shabat system of differential equations associated with the mKdV equation and obtain a representation of the solution of the mKdV equation through Marchenko integral equations of the inverse scattering method. The kernel of these equations is valid only for $x\geq 0$ and it takes into account all specific properties of the pair of compatible differential equations in the chosen half-strip or in the quarter plane. The main result is the collection A-B-C of characteristic properties of the scattering functions given in the paper.

math.AP↗