Semidiscretization in time for nonlinear Schrödinger-waves equations

Thierry Colin; Pierre Fabrie

doi:10.3934/dcds.1998.4.671

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1998, Volume 4, Issue 4: 671-690. Doi: 10.3934/dcds.1998.4.671

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Semidiscretization in time for nonlinear Schrödinger-waves equations

Thierry Colin^1, and
Pierre Fabrie^2,

1.
Mathématiques Appliquées de Bordeaux, CNRS ERS 123 et, Université Bordeaux 1, 351 cours de la libération, 33405 Talence cedex
2.
Université Bordeaux-I, Mathématiques Appliquées, 351 Cours de la Libération, 33405 Talence Cedex

Received: October 31, 1996

Revised: March 31, 1998

Published: September 1998

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Abstract

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.

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Mathematics Subject Classification: 35L05, 35Q55, 65M12.

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