The Research On Solving Method And Stability Of Exact Periodic Solution For Some Nonlinear Evolution Equations

On the basis of summarizing predecessors’ work,we study a method to obtain exact periodic solutions for a series of Zakharov equation,including Klein-Gordon-Zakharov equation and Zakharov-Rubenchik equation.And the properties of period of those peri-odic solutions were also studied in this paper.Meanwhile,we consider(n+1)dimension coupled nonlinear Klein-Gordon equations.We mainly study how to obtain a series of exact periodic solutions and orbital stability of this periodic solution in this system.Firstly,inspired by[1],in combination with Jacobian ellipse function method,we get the exact periodic solutions for a class Zakharov equation,Klein-Gordon-Zakharov equation and Zakharov-Rubenchik equation.Moreover,in a neighborhood of the relevant wave velocity c,we prove that the periods of the periodic solutions of Zakhaxov equation and Klein-Gordon-Zakharov equation are function of wave velocity c,respectively.Next,we can obtain that the period of the periodic solution of Zakharov-Rubenchik equation is also function of wave velocity c.Secondly,we consider(n + 1)dimension coupled nonlinear Klein-Gordon equations.By Jacobian ellipse function method,we acquire the periodic solution of this system.Furthermore,we also prove that the period of the periodic solutions of this system is function of wave velocity c.Lastly,by applying orbital stability theory,which was established by M.Grillakis,J.Shatah and W.Strauss[2],we study orbital stability of this exact periodic solution of(n+1)dimension coupled nonlinear Klein-Gordon equations.Our paper is divided into five chapters.In the first chapter,we mainly introduced the development of nonlinear science and soliton,some significant methods of solving nonlinear evolution equations,historical background and research development stability of nonlinear evolution equations.At last,we give a brief introduction of the main content of this paper.In the second chapter,we state some definitions and some basic facts.In the third chapter,we consider the following equations,Zakharov equation Klein-Gordon-Zakharov equation and Zakharov-Rubenchik equation By Jacobian ellipse function method,we obtain exact periodic solutions of above these equations.Furthermore,in a neighborhood of the wave velocity c,we prove that the peri-ods of the periodic solutions of Zakharov equation and Klein-Gordon-Zakharov equation are the function of wave velocity c,respectively.Next,by a simple analysis method,we can obtain that the period of the periodic solution of Zakharov-Rubenchik equation is the function of wave velocity c.In the fourth chapter,we discuss the following(n+1)dimension coupled nonlinearKlein-Gordon equations At first,we obtain exact periodic solution Moreover,we also prove that the period of the periodic solutions of this system is function of wave velocity c in its neighborhood.Next,we prove that this solution is orbital stable.In the fifth chapter,we summarized the whole paper,and propose some problems for the future research and explore.