Convergence of a numerical scheme for a coupled Schrödinger–KdV system
- 196 Downloads
We prove the convergence in a strong norm of a finite difference semidiscrete scheme approximating a coupled Schrödinger–KdV system on a bounded domain. This system models the interaction of short and long waves. Since the energy estimates available in the continuous case do not carry over to the discrete setting, we rely on a suitably truncated problem which we prove reduces to the original one. We present some numerical examples to illustrate our convergence result.
KeywordsNonlinear Schrödinger equation Korteweg–de Vries equation Short wave long wave interaction Finite difference scheme
Mathematics Subject Classification35Q55 35Q53
The authors were supported by the Portuguese Foundation for Science and Technology (FCT), PEst OE/MAT/UI0209/2011, and the FCT grant PTDC/MAT/110613/2009. PA was also supported by FCT through a Ciência 2008 fellowship.
- 8.Kato, T.: On the Cauchy problem for the (generalized) Korteweg–de Vries equation. Studies in applied mathematics. In: Adv. Math. Suppl. Stud., vol. 8, pp. 93–128. Academic Press, New York (1983) Google Scholar
- 13.Tsutsumi, M.: Well-posedness of the Cauchy problem for a coupled Schrödinger–KdV equation. Nonlinear mathematical problems in industry, II. In: GAKUTO Internat. Ser. Math. Sci. Appl., vol. 2, pp. 513–528, Iwaki, 1992 (1993) Google Scholar