Symmetry, Integrability and Geometry: Methods and Applications (SIGMA)

SIGMA 14 (2018), 119, 19 pages      arXiv:1805.05153      https://doi.org/10.3842/SIGMA.2018.119

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

Rustem R. Aydagulov ab and Alexander A. Minakov cd
a) Lomonosov State University, Leninskie Gory 1, Moscow, Russia
b) A.A. Blagonravov Institute of Mechanical Engineering, Russian Academy of Sciences, Bardina 4, Moscow, Russia
c) International School for Advanced Studies (SISSA), via Bonomea 265, Trieste, Italy
d) Institut de Recherche en Mathématique et Physique (IRMP), Université catholique de Louvain (UCL), Chemin du Cyclotron 2, Louvain-La-Neuve, Belgium

Received May 15, 2018, in final form October 24, 2018; Published online November 07, 2018

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.

Key words: stimulated Raman scattering; Riemann-Hilbert problem; Whitham modulation theory; integrable systems.

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