Study On Exact Solutions And Integrability Of Some Nonlinear Equations

The extensive application of nonlinear science in nonlinear optics,plasma physics,fluid mechanics and other fields of physics has attracted the attention of many mathematicians and physicists.Integrability is an important research topic in nonlinear science,and studying the integrability of nonlinear equations helps people analyze and understand the natural phenomena it reflects.This thesis is based on Hirota bilinear operators and Bell polynomial theory to study the integrability and exact solutions of the(1+1)dimensional variable coefficient complex mKdV equation and the(2+1)dimensional variable coefficient Boussinesq equation.The main contents of this article are as follows:The introduction section mainly introduces the background of the nonlinear equations and the current research status at home and abroad,then introduces the definitions and properties of Hirota bilinear operators and Bell polynomials,and finally introduces the main work of this article.In Chapter 2,Bell polynomials and Hirota bilinear method are used to derive the bilinear form and bilinear B(?)cklund transformation of the(1+1)dimensional variable coefficient complex mKdV equation.Based on the obtained bilinear B(?)cklund transformation,the Hopf-Cole transformation is used to derive the Lax pairs of the(1+1)dimensional variable coefficient complex mKdV equation.In Chapter 3,Bell polynomials and Hirota bilinear method are used to derive the bilinear form of the(2+1)dimensional Boussinesq equation with variable coefficients.The N-soliton solution of the equation is obtained and the evolution and interaction of the soliton solution are analyzed.By studying the bilinear form of the equation,the bilinear B(?)cklund transformation of the equation is derived.Chapter 4 provides a brief summary of this article and plans for future content that is worthy of in-depth consideration and research.