An engineer's guide to soliton phenomena: Application of the finite element method

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 Schrodinger 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.

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