The Soliton Solutions Of Discrete Integrable Systems

The soliton theory who combines the Applied mathematics with Mathemat-ical physics perfectly is an significant branch of the natural science, as its wide applications in practice it draws more and more attentions from the world, and nowadays the research of the soliton theory has penetrated into almost all of the natural sciences.With the discovery of the discrete soliton in Biological system, Photon structure, Atomic chains, etc, to get the exact soliton solutions of the non-linear discrete system becomes one of the focus in the nonlinear science. Nonlinear differential-difference equation is semi-discrete, that is to say that part of or all of the space variables is discrete but the time variable is always continuous.In the paper, the soliton solutions of the nonlinear differential-difference e-quations are mainly discussed.Chapter one is briefly about the discovery and de-velopment of the solitons, the solutions of the nonlinear evolution equations and the overview of the nonlinear differential-difference equation.In the first section of chapter two, the extended Tanh method are systematicly summarized and the idea and the technique of hyperbolic Tanh function method are explored; in the sec-ond section the specific calculational steps and application of the Complex Tanh function is given. In chapter three, the main idea and the application of discrete Ja-cobi sub-equation method are described in details. In chapter four, several discrete integrable systems are presented and obtains the exact solutions of the discrete e-quations with the method just mentioned. And there are the main references and the acknowledgements at the end of the paper.