Research On Exact Solutions Of Nonlinear Equations Based On Darboux Transformation And Bilinear Form

As a member of nonlinear scientific research,soliton theory has played an important role in fields of oceanology,nonlinear optics,electromagnetism,etc..At present,the establishment of integrable systems and the solving to nonlinear equations are two main academic topics in soliton theory,which have attracted close attention from scholars at home and abroad.It is well known that exact solutions to nonlinear equations not only explore essential structures of nonlinear equations in depth,but also help us further understand real physical phenomena.In this thesis,aimed at nonlinear equations,we mainly investigate and improve topical solving methods in soliton theory including Darboux transformation and several solving methods which are derived from the Hirota bilinear method in order to generate abundant exact solutions.The full structure of this paper is as follows:In the first chapter,we demonstrate the original development of soliton,and summarize key research and our main works in soliton theory.In the second chapter,we briefly state the basic idea of Darboux transformation which can be applied to the Dirac-type equation and super NLS-m Kd V equation.In addition,generated solutions are presented graphically.In the third chapter,we extend the transformed rational method and separately derive abundant exact solutions including periodic traveling wave solutions,hyperbolic function solutions,complexiton solutions and resonant multiple wave solutions to a extended Jimbo-Miwa equation.In the fourth chapter,based on the Hirota bilinear forms,lump-type solutions and lump solutions corresponding to(3+1)extended Kadomtsev-Petviashvili equation and its reduced equation when variables z=x are showed by the positive quadratic function method,then the detailed structures and dynamical properties of generated solutions are presented analytically and graphically.In addition,we apply the generalized positive quadratic function method on Hirota-Satsuma-Ito equation and two new research objects so that two kinds of lump-stripe solutions,rogue wave solutions and multiple lump solutions are respectively generated.In the last chapter,we give a concluding remark and prospect for future research.