| Nonlinear evolution equations have received much attention in physics and mathematics.In recent years,different research methods have been used to solve the nonlinear development equation,so as to obtain the exact solution of the nonlinear development equation,including homogeneous equilibrium method,Bell polynomial method,tentative function method and Hirota bilinear method.In this paper,several nonlinear development equations are studied.Based on Hirota bilinear method,Bell polynomial method,trial function method,Painlevé analysis method,homogeneous balance method,B?cklund transformation of nonlinear development equation,Painlevéintegrability,Lax integrability,construction of infinite conservation law and construction of exact solutions are studied.In Chapter 1,the physical background and application methods of nonlinear development equations are introduced.Hirota bilinear method,Bell polynomial method,Painlevé analysis and homogeneous balance method are mainly introduced here.In Chapter 2,Hirota bilinear method is used.(3+1)-dimension B-type Kadomtsev-Petviashvili-Boussinesq(BKPB)equation and(4+1)-dimension variable coefficient Boiti-Leon-Manna-Pempinelli(BLMP)equation are reduced to bilinear form,Then the exact solution of the(3+1)-dimensional BKPB equation and the Lump solution of the(4+1)-dimensional variable coefficient BLMP equation are obtained by trial function method.Finally,the dynamic characteristics of the obtained solutions are analyzed by using the symbolic computation system Mathematica.In Chapter 3,based on the Bell polynomial method,the bilinearizati-on of the(3+1)-dimensional shallow water wave equation,the generaliz-ed variable coefficient(2+1)-dimensional Lax-Kadomtsev Petviashvili(Lax-KP)equation,and the(2+1)-dimensional variable coefficient Nizhnik-Novikov-Veselov(NNV)equation are studied first.Then,the Bell polynomial B?cklund transformation,infinite conservation law,and Lax integrability of these three nonlinear evolution equations are constructed respectively.Finally,with the aid of auxiliary functions,the Lump-Kink solution of the(3+1)-dimensional shallow water wave equation and the single soliton solution,double soliton solution,and N-soliton solution of the generalized variable coefficient(2+1)-dimensional Lax-KP equation are constructed.By selecting appropriate parameters,three-dimensional and contour maps of nonlinear evolution equations are obtained using the symbolic computing system Mathematica.In Chapter 4,we first construct the self B?cklund transformation of the generalized(3+1)-dimensional Kadomtsev-Petviashvili-Boussinesq(K PB)equation and the generalized variable coefficient(3+1)-dimensional Hirota-Bilinear(HB)equation using the homogeneous balance method.Secondly,using the Painlevé analysis method,we prove the integrability of two nonlinear evolution equations in the sense of Painlevé.Finally,through the self B?cklund transformation of the generalized(3+1)-dimensional KPB equation,a block kink solution is constructed.Using the B?cklund transformation of the generalized variable coefficient(3+1)-dimensional HB equation,a higher order strange wave solution is constructed.Summary and outlook of a simple summary of this paper,and look forward to the future worthy of deep thinking of the research content. |
