Painlevé Integrability And Related Problems Of Some High-dimensional Nonlinear Evolution Equations

Taking computer algebra system as a tool,this paper studies the Painlevéintegrability,exact solutions and B?cklund transformation of some nonlinear evolution equations,such as generalized(3+1)-dimensional shallow water wave equation,WBK equations,(4+1)-dimensional variable-coefficient KP equation,(3+1)-dimensional variable-coefficient KP equation and(3+1)-dimensional variable-coefficient STOL equation.Chapter 1 briefly summarizes the physical and biological background,integrability and solution methods of nonlinear evolution equations,which includes Painlevé analysis,Hirota bilinear method,variable separation method.In Chapter 2,the integrability of generalized(3+1)-dimensional shallow water wave equation in the sense of Painlevé is proved by using Painlevéanalysis method;Secondly,according to the truncated Painlevé expansion,the B?cklund transformation between the generalized(3+1)-dimensional shallow water wave equation and the linear equation is constructed,and the variable separation solutions of the generalized(3+1)-dimensional shallow water wave equation are found by the B?cklund transformation;Finally,the hybrid solutions of the generalized(3+1)-dimensional shallow water wave equation are obtained by Hirota bilinear method.In Chapter 3,the integrability of WBK equations in the sense of Painlevéis proved by Painlevé analysis method;Secondly,based on the truncated Painlevé expansion,the B?cklund transformation of WBK equations is constructed;Finally,the superposition solutions of WBK equations are obtained by B?cklund transformation.In Chapter 4,this chapter studies the hybrid solutions,Painlevéintegrability and B?cklund transformation of(4+1)-dimensional variable-coefficient KP equation,(3+1)-dimensional variable-coefficient KP equation and(3+1)-dimensional variable-coefficient STOL equation.(1)The Hirota bilinear method is applied to the(4+1)-dimensional variable-coefficient KP equation,and the bilinear form is obtained.By using the trial function method,several exact solutions are constructed,including Lump solution,Lump soliton solution,Rogue soliton solution,Lump kink solution,E-type Lump solution and E-type Lump soliton solution.(2)Using Painlevé analysis method,the Painlevé integrability of(4+1)-dimensional variable-coefficient KP equation and(3+1)-dimensional variable-coefficient KP equation are discussed respectively.In addition,the B?cklund transformation of(4+1)-dimensional variable-coefficient KP equation and the superposition solutions of different functions are studied.(3)The superposition solutions of(3+1)-dimensional STOL equation with variable-coefficient are studied by using variable separation method.Step 1: Transform the(3+1)-dimensional STOL equation with variable-coefficient into bilinear form.Step 2: Based on the bilinear form of the equation,the superposition solutions of the equation combined by different functions are constructed by variable separation method.Step 3:Draw three-dimensional maps and contour maps with the help of symbolic computing system Mathematica and the arbitrariness of the coefficients of the equation,and analyze the dynamic characteristics of the solutions.