An engineer’s guide to soliton phenomena: Application of the finite element method. (English) Zbl 0624.76020

(Authors’ abstract.) The paper attempts an elementary survey of the physical and mathematical background appertaining to solitons and discusses in particular the numerical solution of three types of dispersive nonlinear partial differential equations exhibiting soliton- type solutions, namely the Korteweg-de Vries equation, the nonlinear Schrödinger equation, and the Sine-Gordon equation. Throughout this study a semidiscrete Galerkin method is applied using a finite element discretization in space and a step-by-step time integration of the resulting system of nonlinear ordinary differential equations. Depending upon the special type of the evolutionary equation the application of a Petrov-Galerkin procedure may increase significantly the numerical stability. Accuracy and effectivity of the different approaches are demonstrated on a series of computer plots.
Reviewer: H.Lange


76B25 Solitary waves for incompressible inviscid fluids
65N30 Finite element, Rayleigh-Ritz and Galerkin methods for boundary value problems involving PDEs
35Q99 Partial differential equations of mathematical physics and other areas of application
76M99 Basic methods in fluid mechanics
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