The linear stability of shock waves for the nonlinear Schr\"odinger-Inviscid Burgers system
Analysis of PDEs
2012-12-11 v3 Numerical Analysis
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.
Keywords
Cite
@article{arxiv.1202.5837,
title = {The linear stability of shock waves for the nonlinear Schr\"odinger-Inviscid Burgers system},
author = {Paulo Amorim and Joao-Paulo Dias and Mario Figueira and Philippe G. LeFloch},
journal= {arXiv preprint arXiv:1202.5837},
year = {2012}
}
Comments
19 pages