Study On The Exact Solutions Of Several Nonlinear Equations

In recent years,people pay more and more attention to the study of nonlinear partial differential equation,which is widely applied to neuroscience,physics,communication,biology and many other disciplines.Seeking exact solutions of the equations is helpful to understand the essential properties,algebraic structure and physical phenomena.In this thesis,the exact solutions of generalized constant coefficients KPB equation and generalized variable coefficients Hirota equation are studied based on Hirota bilinear method and Darboux transform method.In chapter 1,the research background of soliton theory,the research status at home and abroad and the main work of this thesis are introduced.In chapter 2,some preparatory knowledge of Hirota bilinear method and Darboux transformation method are mainly introduced.Such as Bell polynomial theory,bilinear derivative theory,Lax pair,gauge transformation and so on.The Kd V equation is taken as an example to demonstrate the solution ideas of the two methods.In chapter 3,the exact solution of the generalized constant coefficient KPB equation is obtained by using Hirota bilinear method.Based on the theory of Bell polynomials,the bilinear form of the equation is observed by selecting appropriate variable transformations.By assuming the form of the solution and expanding it according to the power,the single soliton solution and double soliton solution of the equation are obtained,and the n-soliton solution is derived recursively.The three-dimensional density distribution diagram,top view and time evolution diagram of single and double solitons are drawn with fixed parameters,and their dynamic behaviors are analyzed.In chapter 4,the exact solution of generalized variable coefficients Hirota equation is obtained by using Darboux transformation method.The classical transform of the equation is constructed based on the Lax pair of the equation.Starting from the zero seed solution,the single-,double-and n-soliton of the equation are obtained by selecting different spectral parameters for several iterations.The three-dimensional density distribution diagram and top view of single and double solitons are drawn with different parameters.The influence of its coefficients on soliton and its dynamic behavior are investigated.In chapter 5,the exact solution of generalized variable coefficients Hirota equation is obtained by using generalized Darboux transformation method.According to the generalized Darboux transform,starting from the non-zero seed solution,the single-,double-and n-rogue wave solutions are obtained by selecting the same spectral parameters for several iterations.The three-dimensional density distribution diagram and top view of single and double rogue waves are drawn with different parameters.The influence of its coefficients on soliton and its dynamic behavior are investigated.