Nonlinear Integrable Systems And Related Topics

The major contents in this dissertation consist of:1.The generalized MKdV equation hierarchy is constrcted from a 3×3 matrix spectral problem,then,its Hamilton structure is presented and the Liouville integrability is demonstrated.By establishing binary symmetric constraints,the constrained flows of the hierarchy are presented,which are then reduced to finite-dimension Hamilton systems. Then,an integrable couplings system is obtained by applying a semi-direct sum of Lie algebras.Then the Hamilton structure of the integrable couplings is constructed by the variational identity.The nonisospectral noncommutative KP hierarchy is presented by pseudo-difference operator technique.2.Frobenius integrable decompositions are introduced for two classes nonlinear evolution equations with variable coefficients for the first time,including the KdV equation and potential KdV equation,the Boussinesq equation and the Camassa-Holm equation with variable coefficients etc.The generalized(2+1)-dimensional KP,cKP and mKP were decomposed into the(1+1)-dimensional integrable equations.We investigate the relations between the consistent solutions of the 2-order and 3-order complex AKNS equations and the solutions of three(2+1)-dimensional soliton equations.With the help of the Darboux transformation,we get the explicit solutions of the generalized KP equation,cKP equation and mKP equation,and they are presented in double Wronskian form.3.The first one is to derive the exact solutions for the fifth-order KdV equation and its constraint equation through Hirota method and Wronskian technique,respectively.Further, the uniformity of these two kinds of exact solutions is proved.The second one is to obtain the general double Wronskian solution of the isospectral Levi equations by generalizing the equation satisfied by Wronskian entries to the matrix equation,including soliton solutions,rational solutions,Matveev solutions,complexiton solutions and interaction solutions. The last part,we derive the double Wronskian solution of the non-isospectral Levi equations.The dynamics including one-soliton characteristics and two-solitons scattering for the isospectral and non-isospectral Levi equations are investingated analytically.