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October  1998, 4(4): 671-690. doi: 10.3934/dcds.1998.4.671

Semidiscretization in time for nonlinear Schrödinger-waves equations

1. 

Mathématiques Appliquées de Bordeaux, CNRS ERS 123 et, Université Bordeaux 1, 351 cours de la libération, 33405 Talence cedex, France

2. 

Université Bordeaux-I, Mathématiques Appliquées, 351 Cours de la Libération, 33405 Talence Cedex

Received  November 1996 Revised  April 1998 Published  July 1998

In this paper, we are concerned with Crank-Nicolson like schemes for:

$ (NLW_\omega ) \frac{1}{\omega^2} \partial_t^2 E_\omega -i\partial_t E_\omega -\D E_\omega =\lambda | E_\omega |^{2\sigma} E_\omega. $

We present two schemes for which we give some convergence results. On of the scheme is dissipative and we describe precisely the dissipation. We prove that the solution of the second scheme fits that of $(NLW_\omega )$ while the first one compute a average value of the solution.

Citation: Thierry Colin, Pierre Fabrie. Semidiscretization in time for nonlinear Schrödinger-waves equations. Discrete & Continuous Dynamical Systems - A, 1998, 4 (4) : 671-690. doi: 10.3934/dcds.1998.4.671
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