We investigate the coupling between the nonlinear Schrödinger 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 Schrödinger–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.
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The authors are grateful to Luis Sanchez for many discussions. The first three authors were partially supported by the Portuguese Foundation for Science and Technology (FCT) through the grant PTDC/MAT/110613/2009 and by PEst OE/MAT/UI0209/2011. The first author (P.A.) was also supported by the FCT through a Ciência 2008 fellowship. The fourth author (PLF) was supported by the Centre National de la Recherche Scientifique (CNRS) and the Agence Nationale de la Recherche through the grants ANR 2006-2–134423 and ANR SIMI-1-003-01.
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Amorim, P., Dias, J., Figueira, M. et al. The Linear Stability of Shock Waves for the Nonlinear Schrödinger–Inviscid Burgers System. J Dyn Diff Equat 25, 49–69 (2013). https://doi.org/10.1007/s10884-012-9283-0
- Schrödinger–Burgers system
- Nonlinear Schrödinger equation
- Shock wave
- Linear stability