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Volume 3,     Number 1,     Winter 1995

 

THE EFFECT OF CHANGE IN THE NONLINEARITY
AND THE DISPERSION RELATION OF
MODEL EQUATIONS FOR LONG WAVES
J.L. BONA AND M. SCIALOM

Abstract. The purpose of this paper is to understand the dependence of solutions of nonlinear, dispersive wave equations on the nonlinearity and the dispersion relation. This program of study is carried out here in the relatively specific, but practically important context of Korteweg-de Vries-type equations. In the last part of the paper, it is shown how the results for the Korteweg-de Vries equation and its relatives may be adapted to other classes of model equations such as nonlinear Schrödinger-type equations and regularized longwave equations. The general thrust of our results is that small perturbations of a given dispersion relation or nonlinearity make only a small difference in the solution over a relatively long time scale. While not unexpected, this kind of theorem is useful as a guide to model builders in showing what sort of approximations can be countenanced without affecting the resulting solutions in an intolerable way.

 

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