Integrability Of Two Kinds Of Soliton Equations Based On Bell Polynomials

The study on soliton equations integrability can well be used to describe the physical phenomena in the fields of communication physics,condensed matter physics,fluid mechanics and ocean engineering.In this thesis,the integrability of two kinds of soliton equations is studied from different angles by using Bell polynomials and related theories and Hirota bilinear method.Firstly,the research background,the research status of the soliton theory at home and abroad and the main work of the thesis are described briefly.Secondly,Hirota bilinear derivative,Bell polynomials and related theories are introduced,and the bilinear method of nonlinear partial differential equations are introduced by concrete examples.The main research contents of this paper are as follows:1.Based on Bell polynomials theories,the integrability of(2+1)dimensional generalized KPB equation is studied.Firstly,based on the bilinear form of the equation,the bilinear B(?)cklund transformation of the equation is obtained under certain restrictive conditions.Then the B(?)cklund transform is linearized to obtain the Lax pair of the equation.Finally,the infinite conservation law of the equation is obtained by substituting the series expansion.2.By using the Hirota bilinear operators and Bell polynomials theories,the integrability of the(3+1)dimensional generalized shallow water wave equation is studied.Firstly,the bilinear form of the equation with functionψ(y,z)about the variablesyandzis obtained.Secondly,the soliton solutions of the equation is obtained by Hirota bilinear method.The images of single soliton,double soliton and triple soliton solutions are obtained by different values of coefficients,and the interactions among solitons is discussed by analyzing soliton solutions images.Then,the bilinear B(?)cklund transformation of the equation is obtained under certain constraints.The Lax pair of the equation is obtained by linearizing the B(?)cklund transformation.Finally,the infinite conservation law of the equation is obtained by substituting the series expansion.