Research On Exact Solutions Of Several Nonlinear Integrable Equations

Theory of integrable system is a crucial part of nonlinear partial differential equation and mathematical physics,whose critical part is research on the inte-grability of an extensive class of nonlinear partial differential equations,nonlinear ordinary differential equations and differential-difference equations.One of the vi-tal research subjects in the theory of integrable system is finding exact solution to nonlinear partial differential equations and analyzing dynamical properties of the solutions.The main work of this dissertation is exactly solving several nonlinear inte-grable systems,which extends the theory of integrable system by valuable new findings.It widely known that inverse scattering method,the Darboux-Backlund transform method and the Hirota bilinear method are three key method in find-ing exact solution.the Hirota bilinear method,which is a vital method in finding exact solution to nonlinear equations,is the main method in this dissertation.Al-though there is a number of reach studies on exactly solving nonlinear equations by the Hirota bilinear method,this method is still one of the important methods in finding exact solutions.A great many of difficult problems lie in this method.Typically,the representation of multi-soliton solution in Pfaffian requires strong technic,which is considerably challenging work.In this dissertation,we discuss a(2+1)-dimensional differential-difference system,which is an integrable discretization of the partial differential equation composed by Maccari from the KP equation.Maccari's system has been studied extensively,such as Lax integrability,Palaneve property,doubly periodic solutions and soliton solutions.We give the N-soliton solution to this system by the Hirota bilinear method.By discretizing the Hirota D operator on the x axis,we compose the integrable semi-discretization of Maccari's system and find the N-soliton solu-tion to the semi-discrete system.We then describe the asymptotic property of the solution,give the relationship between some certain coefficients and the phase shift of soliton interaction and the resonant phenomenon.Numerical scheme based on the semi-discrete system is composed to solve the initial-boundary-value problem of the Maccari's system.Next,we study a generalized M-component AB system.The AB system de-scribes the propagation of wave pulse in a two-level medium.It can also describe the ultra-short pulse in nonlinear optics.As the nonlinear Schrodinger equation,all focusing,defocusing,focusing-defocusing case of the generalized M-component AB system are studied.We compose the N-soliton solutions to the generalized M-component AB system in Pfaffian.Bright-dark soliton solutions in the focus-ing case and the defocusing case,the bright soliton solutions and the dark soliton solutions in the focusing-defocusing case are new properties and attract attention.Then,we generalize the complex coupled integrable dispersionless equation to the complex multi-component case.The complex coupled integrable dispersionless equation describes a current-fed string interacting with an external magnetic field and has close relationship with the sine-Gordon equation.We present the Lax pair and compose the bright N-soliton solution in Pfaffian.In the final chapter,we compose three coupled bilinear equations and give their exact solutions in Pfaffian.