The Study On Multi-Soliton Solutions Of Integrable Systems With Symbolic Computation

In this dissertation,with the help of symbolic computation,the multi-soliton solutions of integrable systems are obtained by the N fold Darboux matrix method,the diagonal Darboux matrix method(DDMM),the Hirota's direct method and the Wronskian technique.The properties of these solutions are also investigated.In addition,the integrable coupling system of the coupled KdV hierarchy and its bi-Hamiltonian structures are constructed by semi-direct sums of Lie algebra and the variational identity.Chapter 1 is the research background related to the dissertation.We briefly outline the development of the theory of solitons and integrable systems.Subsequently,we summarize the recent development and achievement in the theory of solitons and integrable systems at home and abroad.In chapter 2,we first provide a unified explicit form of N fold Darboux transformation for a class of isospectral problem via the N fold Darboux matrix method.Some new bidirectional multipeak N-soliton solutions of some soliton equations associated with the generalized Broer-Kaup-Kupershmidt and Boussinesq-Burgers(BKK-BB)spectral problem are presented by the Darboux transformation.In addition,with the help of the reduction technique and the decomposition technique, we can get a series of N-soliton and N-complexiton solutions of some nonlinear evolution equations via the N fold Darboux transformations.As an applications,some new multi-soliton and multi-complexiton solutions for the complex mKdV equation,a(3+1)-dimensional nonlinear evolution equation and the(2+1)-dimensional KP equation are explicitly given.In chapter 3,the Darboux transformations for a new spectral problem,the Boiti-Tu spectral problem and a generalized Kaup-Newell spectral problem are constructed by the diagonal Darboux matrix method.As an applications,we present N-soliton solutions of a coupled integrable dispersionless equation,a series of solitary wave solutions of the generalized coupled mKdV equation and some new periodic solutions of the generalized derivative nonlinear Schrodinger equation. Based on the diagonal Darboux matrix method,a new efficient algorithm for constructing the Darboux transformation is presented.The algorithm has been implemented with Maple 12.In chapter 4,extending the application of the Hirota's direct method in soliton equations, multi-complexiton solution formulae of bilinear soliton equations are derived by changing the real parameters into conjugated complex parameters in pairs.When the parameters are suitably chosen, we can obtain the N-soliton solutions from the N-complexiton solution formulae.Further,we derive a series of non-singular multi-complexiton solutions of some nonlinear evolution equations.In chapter 5,we present a generalized Wronskian solution formula of a(3+l)-dimensional nonlinear evolution equation,in which positon solutions,negaton solutions,soliton solutions, complexiton solutions and interaction solutions are included.And we compute a double Wronskian solution formula of the(3+1)-dimensional nonlinear evolution equation,by which some rational solutions are obtained. In chapter 6,we derive an integrable coupling system of the coupled KdV hierarchy by semidirect sums of Lie algebra.Based on the variational identity,we further construct bi-Hamiltonian structures of the integrable coupling system.The N fold Darboux matrix method is applied to the integrable coupling system associated with the coupled KdV hierarchy for the first time.Then N fold Darboux transformation of the integrable coupling system is constructed successfully.