Application Of Darboux Transformation To Some Soliton Equations

Darboux transformation is a more direct and effective method to study the exact solutions of soliton equations.It establishes the relation between two different solutions of the same equation.Thereby,we can obtain many nontrivial solutions from trivial solutions of the equation.The aim of the present paper is to discuss the Darboux transformation and its application of two soliton equations,which are associated with two 3×3 matrix spectral problems.In the first place,we introduce the soliton theory and background of Darboux transformation.Secondly,resorting to the gauge transformation of the corresponding 3×3 matrix spectral problems,we derive a first-order and N-order Darboux transformation for the generalized TD equation.As an application,we get two nontrivial exact solutions from two trivial initial solutions.At last,we consider the Darboux transformation and exact solutions of a differential-integral equation.