| The major contents in this paper include: the exact solutions of the nonlinear evolution equations and the extended integrable models of which are Hamiltonian form. In the second chapter, firstly the exact solitary wave solutions of some equations are obtained according to the homogeneous balance principle, these equations include: the 1+1 dimensional generalized KPP equation and its reduced equations,Huxley equation,the 2+1 dimensional Boussinesq equation,the 2+1 dimensional dispersive long wave equation, simultaneously, we presented the figures and the viewprogram of the exact solitary wave solutions by use of Matlab. Secondly, we find that the Backlund transform of the anamorphic Boussinesq equation System II and the generalized column Kadomtsev-Petviashvilli(KP) equation, in addition, the new solutions of them are constructed from the ready solutions by virtue of the obtained Backlund transform. Thirdly, by use of Jacobi elliptic function method and its extended method, abundant periodic solutions are obtained for the RLW-Burgers equation,the modified Kawachara equation,the five degree Ito-mKdV equation,the Konopelchenko-Dubrovsky(KD) equation, and the figure of parts of the solutions are presented by writing Matlab program. In the third chapter, we firstly illuminate that the expanding integrable models are of Hamilton form, which are constructed from matrix loop algebra by use of On-triangle semi-direct sums method, also, we prove that the result can be used to the integrable hierarchies with the spectral of vector form or multi-component form, and the corresponding expanding integrable hierarchies are of Hamiltonian form. As their applications, we search for the expanding models and their Hamiltonian form of the AKNS hierarchy, Tu hierarchy, multi-component M-AKNS-KN hierarchy by means of the generalized Lie algebra semi-direct sums method, eventually, we testify that the expanding integrable models are also Liouville integrable. |
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