Study On The Composite Wave Solutions Of Some Nonlinear Evolution Equations

Based on the Hirota bilinear method,Bell polynomial method,trial function method and auxiliary equation method,this paper studies the bilinearization,integrability,double Bell polynomial B(?)cklund transformation,bilinear B(?)cklund transformation,exact solution and construction of infinite conservation laws of several high-dimensional nonlinear evolution equations with the help of computer algebra system.New results have been obtained.The first chapter briefly introduces the research background,research significance and research methods of soliton theory.Such as Hirota bilinear method,auxiliary equation method,trial function method and Bell polynomial method and their development history.In the second chapter,based on the Hirota bilinear method,the(3+1)-dimensional generalized Kd V equation is first transformed into a bilinear differential equation.Using the trial function method,the lump solution,kink solution,rational solution and breath solution of the equation are constructed.Then,the(3+1)-dimensional shallow water wave equation is transformed into(2+1)-dimensional nonlinear evolution equation by function transformation.On this basis,the Hirota bilinear method is used to transform into bilinear differential equation.Thirdly,using the auxiliary equation method,the lump solution,lump kink solution and soliton solution of the equation are constructed.Finally,the interaction of solutions is analyzed by image analysis.In chapter three,we study the double B(?)cklund transformation,infinite conservation laws and integrability of three(3+1)-dimensional nonlinear evolution equations with variable coefficients.Firstly,by using the Bell polynomial method,the double Bell polynomial B(?)cklund transformation,bilinear B(?)cklund transformation,Lax pair and infinite conservation laws of(3+1)-dimensional variable coefficient NLEE equation,(3+1)-dimensional variable coefficient Potential YTSF and(3+1)-dimensional variable coefficient nonlinear evolution equation are constructed.At the same time,the integrability of these three(3+1)-dimensional variable coefficient nonlinear evolution equations in the sense of Lax is also proved.Then,through the bilinear B(?)cklund transformation,the exact solutions of these three(3+1)-dimensional nonlinear evolution equations with variable coefficients are constructed.Finally,by selecting different parameters,the dynamic behavior of these solutions is analyzed.In chapter four,we study the bilinear B(?)cklund transformation,infinite conservation law and integrability of two(4+1)-dimensional nonlinear evolution equations with variable coefficients.Firstly,by using the Bell polynomial method,the double Bell polynomial B(?)cklund transformation,bilinear B(?)cklund,Lax pair and infinite conservation law of the generalized(4+1)-dimensional variable coefficient Fokas equation and the generalized(4+1)-dimensional variable coefficient CBS equation are constructed.At the same time,the integrability of the two(4+1)-dimensional variable coefficient nonlinear evolution equations in the sense of Lax is also proved.Then,through the bilinear B(?)cklund transformation,the exact solutions of these two(4+1)-dimensional nonlinear evolution equations with variable coefficients are constructed.Finally,by selecting different parameters,the dynamic behavior of these solutions is analyzed.Summary and prospect,is a brief summary of the content of the study,and future research put forward areas worthy of further consideration.