The linear stability of shock waves for the nonlinear Schr\"odinger-Inviscid Burgers system

TL;DR

This paper studies the linear stability of shock waves in a coupled nonlinear Schr"odinger and inviscid Burgers system, providing a stability theorem and numerical validation for shock wave solutions.

Contribution

It establishes the first linearized stability theorem for the Schr"odinger--Burgers system with shock waves, using energy estimates and hyperbolic equation properties.

Findings

Linearized stability theorem proven for shock waves

Numerical experiments support theoretical results

Analysis based on energy estimates and hyperbolic properties

Abstract

We investigate the coupling between the nonlinear Schr\"odinger equation and the inviscid Burgers equation, a system which models interactions between short and long waves, for instance in fluids. Well-posedness for the associated Cauchy problem remains a difficult open problem, and we tackle it here via a linearization technique. Namely, we establish a linearized stability theorem for the Schr\"odinger--Burgers system, when the reference solution is an entropy--satisfying shock wave to Burgers equation. Our proof is based on suitable energy estimates and on properties of hyperbolic equations with discontinuous coefficients. Numerical experiments support and expand our theoretical results.