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

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→+∞, 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 were 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.