The Exact Solutions And Integrable Coupling System Of Nonlinear Equations

The main contents of the paper are:a uniform construction of soliton equations is obtained and its properties are discussed,which are also improved.What is more,the infinite conservation law of KN equation and its Hamiltonian structures are obtained.At last,based on the idea and importance of Darboux transformations,the solutions of AKNS equations are solved.Chapter 1 is devoted to reviewing the history and the development of he soliton theory,solving nonlinear equation and Darboux transformations.At the same time ,the applications of these methods are presented.Chapter 2 mainly introduces the uniform constructions of Kdv equation and 2+1dimension sin-Gordon equations,which are based on the achievements of the presented dissertation.Chapter 3 first presents the integrablity of infinite dynamical system.That is to say,the equations in the system can be expressed as Hamiltonian equations.And there are n independent conservation elements.Some similar properties were found in many infinite dimensional Lax integrable system.In the paper,the infinite conservation law of a Riccati equation was given through the KN equation.And then obtains its Hamiltonian structure.Chapter 4 introduces mainly about the applications of Darboux transformations in the solving nonlinear equations and a discrete problem,AKNS equation,which was enlarged another new form.