Initial-Boundary Value Problem for Stimulated Raman Scattering Model: Solvability of Whitham Type System of Equations Arising in Long-Time Asymptotic Analysis

TL;DR

This paper proves the existence of solutions to the Whitham type equations that describe the long-time asymptotic behavior of a stimulated Raman scattering model, confirming previous assumptions and completing the analysis of the solution's asymptotics.

Contribution

It establishes the solvability of the Whitham type system, thereby justifying the asymptotic analysis of the stimulated Raman scattering model.

Findings

Confirmed the existence of parameters in the Whitham system.

Validated the long-time asymptotic regions of the solution.

Provided mathematical justification for previous asymptotic results.

Abstract

An initial-boundary value problem for a model of stimulated Raman scattering was considered in [Moskovchenko E.A., Kotlyarov V.P., J. Phys. A: Math. Theor. 43 (2010), 055205, 31 pages]. The authors showed that in the long-time range $t\to+\infty$ the $x>0$, $t>0$ quarter plane is divided into 3 regions with qualitatively different asymptotic behavior of the solution: a region of a finite amplitude plane wave, a modulated elliptic wave region and a vanishing dispersive wave region. The asymptotics in the modulated elliptic region was studied under an implicit assumption of the solvability of the corresponding Whitham type equations. Here we establish the existence of these parameters, and thus justify the results by Moskovchenko and Kotlyarov.