A nonlinear model describing a short wave long wave interaction in a viscoelastic medium

TL;DR

This paper introduces a coupled nonlinear Schrödinger and viscoelasticity system modeling short-long wave interactions in media like plasmas or polymers, proving solution existence and uniqueness, and supporting findings with numerical simulations.

Contribution

It extends previous work by establishing local and global solutions for a new coupled nonlinear system involving viscoelastic media.

Findings

Proved existence and uniqueness of solutions

Extended previous results on viscoelastic systems

Provided numerical illustrations of the model

Abstract

In this paper we introduce a system coupling a nonlinear Schr\"odinger equation with a system of viscoelasticity, modeling the interaction between short and long waves, acting for instance on media like plasmas or polymers. We prove the existence and uniqueness of local (in time) strong solutions and the existence of global weak solutions for the corresponding Cauchy problem. In particular we extend previous results in [Nohel \emph{et. al.}, Commun. Part. Diff. Eq., 13 (1988)] for the quasilinear system of viscoelasticity. We finish with some numerical computations to illustrate our results.