Integrability Of Bell Polynomials And Nonlinear Evolution Equations And Related Problems

In this paper,based on Hirota bilinear method and bell polynomial theory,we study the integrability,B(?)cklund transformation and conservation law of several high-dimensional nonlinear evolution equations with the help of computer algebra system,and get new results.Through Hirota bilinear method and homoclinic test method,different kinds of new exact solutions are constructed,and their propagation and evolution characteristics are analyzed.The motion trajectory and physical meaning of the solutions are analyzed by using images.At the same time,according to the theory of bell polynomials,the integrability,B(?)cklund transformation and infinite conservation law of high-dimensional nonlinear evolution equations are studied,and the theorems,corollaries and proofs of solutions of superposition of different functions are given.Studying the superposition solutions of different functions is helpful to understand some important physical phenomena in nonlinear science.In the first chapter,the research background,significance and methods of soliton theory are briefly introduced,such as Hirota bilinear method,bell polynomial and so on.In the second chapter,by using the Hirota bilinear method,the(3+1)-dimensional generalized KdV-type equation is transformed into bilinear form,and we construct the N-soliton solution,lump solution,lump-kink solution,double-kink wave solution and breather solution of the equation.Then,we get the bilinear form and B(?)cklund transformation of(3+1)-dimensional nonlinear evolution equation are constructed,and the higher-order lump solutions,the N-M superposition solutions of higher-order lump solitons and the periodic superposition solutions.Finally,the interaction between the solutions of the two equations is analyzed by using the image analysis method.In chapter three,we study the integrability of(4+1)-dimensional KdV-like equations.On one hand,the bilinear B(?)cklund transformation,lax pair and infinite conservation law of the(4+1)-dimensional KdV-like equation are constructed due to bell polynomial approach,and it is proved that the equation is integrable in lax sense.Then,based on Hirota bilinear method and homoclinic test method,several new exact solutions are obtained,including higher-order lump solutions,higher-order lump kink N-soliton solutions,higher-order lump-cosh-N-cos-M superposition solutions and superposition solutions of different functions.On the other hand,the problem of constructing new exact solutions of(4+1)-dimensional BLMP equation is studied.Firstly,a theorem and its proof for constructing new exact solutions of(4+1)-dimensional BLMP equation are given.Then,different types of solutions of the equation are obtained due to the theorem,moreover,the lump-kink wave solution and the lump-solitary wave solution are obtained.Finally,with the help of the bilinear form of the equation,we get the periodic superposition solution and the compound superposition solution,and the dynamic behavior of these solutions is analyzed by selecting different parameters.In chapter four,we study the interaction between the solution and the solution of three kinds of high dimensional evolution equations with variable coefficients.Firstly,according to Cole-Hopf transformation with non-zero seed solution and trial function method,the breathing kink wave solution,strange wave solution and three solitary wave solution of(3+1)-dimensional DJKM equation with variable coefficients are constructed.Then,based on Hirota bilinear method and homoclinic test method,the theorems,we give corollaries and proofs for constructing new exact solutions of(3+1)-dimensional variable coefficient BLMP equation and(2+1)-dimensional variable coefficient BLMP-BK equation.In addition,new solutions of(3+1)-dimensional variable coefficient BLMP equation and(2+1)-dimensional variable coefficient BLMP-BK equation superposed by different functions are obtained by using the theorem.Finally,due to the arbitrariness of any function in the solution,we choose different functions,and the dynamic behaviors of these solutions are analyzed through three-dimensional and contour plots.In the summary and prospect,we make a simple summary and plan the content worthy of in-depth thinking and research in the future.