| Based on symbolic computation software Maple,and by utilizing symmetry theory,the consistent Riccati equation expansion method,the direct method of constructing op-tion system and the extension theory of higher order symmetries,some related problems of nonlinear models in mathematical physics are investigated.The paper includes three aspects:the construction of nonlocal symmetries and exact solutions of nonlinear differ-ential equations;the classification of option system;the classification of dark equation and construction of recursion operators.The details of the thesis are arranged as follows:In chapter 1,an introduction is devoted to the research background and the curren-t situation related to this dissertation,which includes symmetry theory,option system,dark equations and symbolic computation.The main works of the dissertation are also illustrated.In chapter 2,based on Darboux transformation and the truncated Painleve expansion approach,nonlocal symmetries and residue symmetries of Kaup-Kupershmidt(KK)equa-tion are proposed.Then the nonlocal symmetries are localized by introducing auxiliary variables.By applying the classical Lie symmetry theory to these prolonged systems,the solitons-cnoidal wave interaction solutions are derived by the finite transformation and similarity reduction.Furthermore,some new exact solutions of the AB systems are obtained in virtue of the consistent Riccati equation expansion method.In chapter 3,based on the direct algorithm of one-dimensional optimal system,we study one-dimensional optimal classification of Lie symmetry group of the 2+1-dimensional Wu-Zhang(WZ)equation,and propose the optimal system of WZ equation.By similarity reduction,fifteen classes of complete and inequivalent 1 + 1-dimensional symmetry reduced systems are obtained.It is noteworthy that a new Painleve integrable equation with constant coefficient is contained besides the classic Boussinesq equation and the steady case of the WZ equation.In chapter 4,based on the definition and classification of dark KdV equations by Kupershmidt,firstly,a complete scalar classification for dark modified KdV(MKdV)systems is obtained by requiring the existence of higher order differential polynomial symmetries.Different to the nine classes of the dark KdV cases,there exist twelve in-dependent classes of the dark MKdV equations.Furthermore,for the every class of dark MKdV system,there is a free parameter.Only for a fixed parameter,nine of the dark MKdV can be related to the responding dark KdV via suitable Miura transformation.Besides,the rest three classes of dark MKdV can not be obtained by taking Miura trans-formation of dark KdV cases.Secondly,we generalize the definition of dark equation by the linear homogeneous extensions,the generalized dark MKdV equation with more free parameters are proposed.Taking the appropriate value of some parameters,the general-ized dark MKdV equation can be reduced to the total homogeneous results.The recursion operators of two classes of dark MKdV systems and the generalized dark MKdV systems are given respectively.In chapter 5,the summary and discussion of this dissertation are given,and the outlook of future works are discussed. |
