Symmetry Reductions And Exact Solutions Of Several Partial Differential Equation(s)

Based on the urgent need for solving the nonlinear partial differential equation(s)(PDEs)and the application of symmetry in nonlinear PDEs,this issue aims at some nonlinear PDEs with physical significance and realistic background.The investigation of symmetry,symmetry reduction and classification,and exact solutions for solving PDEs is presented in this paper by the symbolic computation Maple software.By combining the auxiliary functions method with the homogeneous balance method,the various exact solutions of the variant Boussinesq equations(vBEs),the(3+1)-dimensional generalized KP equation((3+1)gKPe),the new(3+1)-dimensional generalized KP equation((3+1)ngKPe),are obtained by applying the Maple software,including trigonometric function solutions,Jacobian elliptic function solutions and rational solutions.Meanwhile,the paper makes use of the Maple to show the pictures of several exact solutions.In order to obtain other transformations of the independent variable,the modified Clarkson and Kruskal method and the classical Lie group method are respectively used to perform detailed analysis on the(3+1)gKPe,the(3+1)ngKPe,the vBEs and the variable Boussinesq-Burgers equations(vBBes).And the paper applies the symmetries of the vBEs to reduce the original equations.The main results are as follows:Chapetr 1 mainly introduces the international research situations on the solutions nonlinear PDEs,and provides introducing background and related basic theory knowledge.Chapetr 2 states that the paper translates the(3+1)gKPe and the(3+1)ngKPe into the ordinary differential equation(ODE)by combining the auxiliary functions method with the homogeneous balance method.The various exact solutions of them are shown by the Maple software,including trigonometric function solutions,Jacobian elliptic function solutions and so on.Meanwhile,the symmetries and the group-invariant solution theorem of them are obtained by the classical Lie group method.Chapetr 3 investigates that the vBEs are transferred into the ODEs by combining the auxiliary functions method with the homogeneous balance method.And the paper gets some solutions,including Jacobian elliptic function solutions etc.What's more,This paper has one-dimensional subalgebra optimal system of the vBEs,which contributes to reduce the original equations to get Painlev?e equations.Chapetr 4 studies on the symmetry classifications for the vBBes.Chapter 5 summarizes the content of the paper and looks forward to what to do in the future.