The Integrable Properties And Solutions Of Several Types Of Nonlinear Models

With the development of modern science and technology,a great quantity of nonlinear models have been put forward in more and more research fields.These nonlinear models are used to describe the phenomena,in the typhoon,tsunami,mining,fluid,optics,communications,biology.We can get more information about the phenomena and learn the essence of some class of phenomena by solving these nonlinear models.In order to solve the nonlinear models,the models can be divided into two types:integrable nonlinear models and non integrable nonlinear models.If the nonlinear models are integrable and have the same integrabe properties,we can solve them by the specified method according to the integrabe properties.By using these method,greate convenience will be brought in the calculation and solving process.If the nonlinear models are non integrable,we haven't appropriate methods to solve because they are too complicated.We can solve them by other methods,for example auxiliary function method,differential substitution method.This thesis is arranged as follows:Chapter one,we introduce the background of the nonlinear models and the development of the exact solutions research work and recent progresses.Chapter two,we introduce the importance of the exact solutions,especially the soliton solutions and its characteristic firstly.Then we provide some methods get the exact solutions,like the inverse scattering method,the homogeneous balance method,Riccati mapping approach,Jacobi elliptic function expansion method,exp expansion approach.Last we introduce Inverse scattering integral,Painleve property,Liouville Integrability and Lax Integrability.The details of the the inverse scattering method's steps is given.Chapter three,we introduce the basic idea of the auxiliary function method and its specific solving steps.Then the exact solutions of Benjamin-Bona-Mahonye equation,Burgers equation and Zakharov-Kuznetsov equation are obtained by using this method.There are not only the rational solutions,but also some soliton solutions.The soliton solutions can show the shape and the amplitude of the solitary wave.We give the pictures of them.Chapter four,firstly we introduce the Painleve truncation expansion method,Conte expansion mthod and the generalized truncated Painleve expansion method.Then we use the WTC method of Painleve analysis and the generalized truncated Painleve expansion mehtod to test the integrable properties of the Burgers which is preseted in section three.Chapter five,we review the entire article and make prospect for the future work.