Solving Nonlinear Partial Differential Equation And Integrable System

In this dissertation, we mainly consider the problems offinding the exact solutions of partial di?erential equations(PDE) inthe theory of soliton and how to get the new integrable couplings inintegrable system:1.the history and development of the soliton theory2.the improved Riccati equation approach for solving PDE3.the auxiliary equation method for solving PDE4.the di?erential manifold and B¨acklund transformation5.Darboux transformation6.integrable couplingsChapter 1 is to introduce the history and development of the soli-ton theory, mathematics mechanization and applications of symboliccomputation, the development of the exact solutions of nonlinear evo-lution equations. As well as the works and achievements that havebeen obtained at home and abroad are listed.Based on the idea of algebraic method,algorithm realization andmechanization for solving nonlinear evolution equations, in Chapter 2,we consider the construction of exact solutions for nonlinear evolutionequations .Firstly we present the improved Riccati equation approachand get some new solutions of the Konopelchenko-Dubrovsky equa-tion.Then the auxiliary equation method is improved to study the anyorder Li′enard equation and many explicit exact solutions of it areobtained.In Chapter 3, with the study of sin-Gordon equation firstly we give the relation of the di?erential manifold and B¨acklund transformation.There is one-to-one mapping of the sin-Gordon equation's solution andpseudo-sphere.Then we present a new Darboux transformation of themulti-parameters coupled GMNLS equations which are the generalizedmultivariable vector form of the Schro¨dinger equation.By applying theDarboux transformation,we obtain new soliton solutions of the coupledGMNLS equations.Chapter 4 is mainly focused on integrable system in soliton equa-tions.For getting the integrable couplings,Professor Gu Fukui gave animportant technique which should satisfy two terms.Here we get rid ofone of them and a special subalgebra G|- of the loop algebra (A|-)₃ is con-structed directly,so that two subalgebras (G|-)₁ and (G|-)₂ of loop G|- meet therelation G|-=(G|-)₁⊕(G|-)₂.By making use of the (G|-)₁,an isospectral problemis established.Again by the use of Tu scheme,a new integrable hier-archy of soliton equations with bi-Hamiltonian structure is obtained.Second,by employing a proper linear combination of the basis of theloop algebra (G|-)₁,another loop algebra (G|=)₁ is presented.It follows thatthe second type of new integrable system is given via Tu scheme.