Study On Darboux Transformation And Conservation Laws Of Three Soliton Equations

Darboux transformation is a direct and effective method to solve exact solutions of soliton equations, it plays an important role in soliton theory. The essence of Darboux transformation is a guage transformation of spectral parameters and it transforms the spectral problem into another spectral problem of the same type. In the process of solving the exact solutions of soliton equations through Darboux transformation, we can use mathematica software combined with symbolic computation to prove partial expressions and plot the graphics of the exact solutions. Obtaining exact solitons of soliton equations through Darboux transformation help us to have a deeper understanding of the solutions of soliton equations and lax pairs for the corresponding spectral problem, therefore, it gets the favour of many scholars. Generaliztion and application of Darboux transformation to more types of soliton equations has gradually developed into an important research direction in the field of soliton theory, having theoretical significance and application value.The main work in this paper can be summarized as: Firstly, in the first chapter, the discovery and development situation of soliton and several methods solving soliton equations were introduced briefly in the Introduction part. Secondly, in the second chapter, we gave the one-fold Darboux transformation for a new differential-difference equation, and obtain new solutions through the known seed solutions. By making use of mathematical software, we drew the graphics of the solutions. Thirdly, in the third chapter, combined with the discrete spectral problem, an infinite number of conservation laws of two differential-difference equations were obtained. Finally, in the fourth chapter, we firstly introduced the third kind of the one-fold Darboux transformation for the generalized coupled Broer-Kaup equation. Then we got the relationship among the three kinds of the one-fold Darboux transformations through the comparison between the two kinds of known Darboux transformations and the third kind of Darboux transformation which we got in front of this chapter. Also the third kind of the one-fold Darboux transformation for the generalized coupled Broer-Kaup equation was generalized to N-fold Darboux transformation. As one application of the obtained Darboux transformation, some new exact solutions were obtained. Besides, the graphics of solutions of the generalized coupled Broer- Kaup are plotted via software.