Study On Various Exact Solutions Of Some Nonlinear Evolution Equations Based On Symbolic Computation

As an important part of nonlinear partial differential equations(NLPDEs),nonlinear evolution equations(groups)(NLEEs)are often used to describe processes that evolve over time.The research objects of this kind of equation come from many fields,such as: life science,information science,chemistry,physics,etc.For specific NLEEs,if the exact analytical solution can be obtained,it will provide great help for people to understand the law of natural phenomena and accurately explain nonlinear phenomena in nature.In recent years,with the rapid development of symbolic computation,the exact solution of NLEEs has gradually become an important research field,and many effective methods for exact solution have been presented.Each of these methods has its own characteristics,but also has internal relations.Compared with the inverse scattering method,the Hirota bilinear method is regarded as a direct method to solve NLEEs.The advantage of this method is that it is an algebraic rather than analytical method.In addition,bilinear neural network method(BNNM)is a new method that combines the Hirota bilinear method and neural network model to obtain the exact analytical solution of nonlinear systems.The application of neural network model in solving NLEEs has attracted the attention of researchers because of its strong nonlinear characteristics.This paper mainly uses bilinear method and bilinear neural network method to solve many exact analytical solutions of NLEEs,such as rogue wave solution,periodic wave solution,breather solution,bright-dark soliton solution and interaction solution,and further analyzes its geometric morphology,physical significance and dynamic characteristics by means of graphs.Specific research contents are as follows:In Chapter 1,the Hirota bilinear method and bilinear neural network method are briefly introduced,and the research and development of rogue wave solution,periodic wave solution,breather solution,bright-dark soliton solution and interaction solution are expound.In Chapter 2,based on bilinear neural network method,using symbolic calculation software Maple,we obtain the rogue wave solutions of(2 + 1)dimensional combined soliton equation and(1 + 1)dimensional Benjamin-Ono equation,including 1-rogue wave solution,3-rogue wave solution and 6-rogue wave solution,and through graphical analysis to understand the dynamics of the form.In Chapter 3,the bright-dark soliton solutions and breather solutions of(2 + 1)dimensional Ito equation and(3 + 1)dimensional B-type Kadomtsev-PetviashviliBoussinesq equation(BKP-Boussinesq equation)are studied by bilinear neural network method.For(2 + 1)dimensional Ito equation,the [3-2-3] model is established,and the bright-dark soliton solutions and breather solutions of the equation are constructed with the help of specific activation functions and corresponding tensor formulas.For(3 + 1)dimensional BKP-Boussinesq equation,the [4-2-3] and [4-3-3] models are established,respectively.The bright-dark soliton solutions and breather solutions of the equation are obtained by assuming specific activation functions.Finally,the trajectories and trends of these solutions in three dimensional space are given.In Chapter 4,based on bilinear neural network method,we present several interaction solutions of(3 + 1)dimensional Hirota bilinear equation(HB equation),(2 + 1)dimensional Ito equation and(3 + 1)dimensional BKP-Boussinesq equation by constructing neural network model and corresponding tensor formula.In particular,we obtain the fractal soliton solution and periodic wave solution for(2 + 1)dimensional Ito equation,and obtain periodic wave solution and rogue wave solution for BKPBoussinesq equation.Finally,the physical meanings of these solutions are further analyzed by means of graphs.In Chapter 5,the work of the full text is summarized,the difficulties encountered in the process of using computer symbol calculation and research is putted as well as the future research work is prospected.