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Numerical simulation of a nonlinear wave equation and recurrence of initial states Buckley, Albert Grant

Abstract

In 1955 Fermi, Pasta and Ulam (FPU) [7] observed an unusual recurrence to initial state in numerical solutions of a nonlinear wave equation. Zabusky and Kruskal (ZK) [47] have subsequently found an explanation for this phenomenon based on special travelling wave solutions ("solitons") of the (nonlinear) Korteweg de Vries (KdV) equation. In this thesis we extend ZK's explanation to a similar nonlinear wave equation given by Johnson[14]. We investigate existence and uniqueness of solitons for a (nonlinear) generalization of the KdV equation. (Chapter II) and present computational results to illustrate ZK's soliton explanation of the recurrence, both for FPU's equation and Johnson's equation (Chapter III). In Chapter IV we give some results concerning the stability of the difference schemes used to obtain solutions to the nonlinear partial differential equations.

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