| Based on symbolic computation, this dissertation investigates the theory with applicationof the soliton and integrable systems of nonlinear mathematical physics. The main work iscarried out in three aspects: based on the Conte expansion Verifcation method, optimizingthis method, and then the truncated invariant expansion method is summarized up to obtainsolutions of non-linear physics models; generalized Korteweg-de Vries (KdV) equation withffth-order dispersion term and singularly perturbed Boussinesq equation are took as examples,showing how to use this method to obtain the approximate solutions and even the exact solutions;spreading the application of this method, using it to solve other nonlinear partial diferentialequations.Chapter1is an introduction to review the theoretical background and development ofsoliton theory, integrable system, nonlinear systems’mathematical research means and symboliccomputation. The main works of this dissertation are also illustrated.Chapter2is the inheritance and development of the theory of chapter1. Based onthe Painlevé analysis method, the truncated invariant expansion method is summarized up tosolve nonlinear partial diferential equations. Three kinds approximate solutions of generalizedKorteweg-de Vries (KdV) equation with ffth-order dispersion term and two kinds approximatesolutions of singularly perturbed Boussinesq equation are obtained by this method. Dealing withthese approximate solutions accordingly, some kinds exact solutions are obtained.Chapter3is the promotion and application of chapter2. The truncated invariant expan-sion method is used to solve generalized Korteweg-de Vries (KdV)equation with fourth-orderdispersion term, KdV-Burgers equation and Burgers equation, their approximate solutions andexact solutions of KdV-Burgers equation and Burgers equation are acquired.Chapter4focuses on the summary and discussion of the whole dissertation, and theprospect for the future work is also put forward. |
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