The Linear Stability of Shock Waves for the Nonlinear Schrödinger–Inviscid Burgers System
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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.
KeywordsSchrödinger–Burgers system Nonlinear Schrödinger equation Shock wave Linear stability
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.
- 1.Amorim, P., Dias, J.-P.: A nonlinear model describing a short wave-long wave interaction in a viscoelastic medium, Q. Appl. Math. doi: 10.1090/S0033-569X-2012-01298-4
- 4.Amorim, P., Figueira, M.: Convergence of a numerical scheme for a coupled Schrödinger-KdV system. Rev. Math. Complut. doi: 10.1007/s13163-012-0097-8
- 9.Cazenave, T.: Semilinear Schrödinger equations, Courant lecture notes in mathematics, vol. 10. American Mathematical Society, Providence (2003)Google Scholar
- 15.Dias, J.-P., Figueira, M., Oliveira, F.: Existence of local strong solutions for a quasilinear Benney system. C. R. Math. Acad. Sci. Paris Ser. I 344, 493–496 (2007)Google Scholar
- 16.Godlewski, E., Raviart, P.-A.: An introduction to the linearized stability of solutions of nonlinear hyperbolic systems of conservation laws. Lecture notes, Lisbon summer school. Ellipse, Lisbon (1999)Google Scholar
- 17.Hörmander, L.: The analysis of linear partial differential operators. I. Distribution theory and Fourier analysis, Grundlehren der Mathematischen Wissenschaften. Springer, Berlin (1983)Google Scholar
- 18.Kato, T.: Linear evolution equations of “hyperbolic” type. J. Fac. Sci. Univ. Tokyo Sect. I(17), 241–258 (1970)Google Scholar
- 19.Kato, T.: Quasi-linear equations of evolution, with applications to partial differential equations, lecture notes in mathematics. Springer, Berlin (1975)Google Scholar
- 20.LeFloch, P.G.: An existence and uniqueness result for two nonstrictly hyperbolic systems, IMA volumes in mathematics and its applications. In: Keyfitz, B.L., Shearer, M. (eds.) Nonlinear evolution equations that change type, pp. 126–138. Springer, New York (1990)Google Scholar
- 21.LeFloch, P.G.: Hyperbolic systems of conservation laws. The theory of classical and nonclassical shock waves, lectures in mathematics. Birkhäuser, Basel (2002)Google Scholar