Darboux Transformation And Explic Solutions Of Soliton Equations

Darboux transformation is a gauge transformation of spectrum problems in the nonlinear equations, and is also one of the very effective methods to get solutions of soliton equations. Darboux transformation can be used to get one soliton solution﹑two soliton solution, and n soliton solution of soliton equations. In order to get a Darboux transformation, we can choose appropriate parameters. Then it reduces the computational complexity of solving the solutions, and gets a lot of valuable solutions. With the classical theory, we can connect Darboux transformation with the Lax pairs of nonlinear systems. By Crum theory, we can directly display second and third Darboux transformation, etc.(u,f) ®(u￠,f ￠) ®(u￠,f￠), so we can get n soliton solution from one soliton solution.There are many kinds of methods to get Darboux transformation. For the soliton equation, by which method to change the equation, this will be determined by the equation of concrete form. For soliton equations with high order Lax matrix, the research on the Darboux transformation is very less, mainly because of its complexity. In the process of getting this kind equation’s transfomation, we need to try several times.In this paper, we consider three important soliton equations:In section one, we introduce Darboux transformation theory and the initial Darboux transformation method. Based on these methods, in the following paper, Darboux transformations of three nonlinear equations are constructed.In section two, we consider an important variable-coefficient soliton equation. By spectral problem of the column kdv, we derive the Darboux transformation, and then get the single soliton solution by means of variation of constants. With the known solution as seed, we have the double soliton solution and n soliton solution by using Darboux theorem and the crum theorem.In section three, we consider Boussinesq-Burgers system. First, according to the zero curature equation, we derivate the Boussinesq-Burgers system. Then, by its spectral problem, We present its basic Darboux transformation. Together with another two basic Darboux transformation, we obtian relationship among the three DT. Then using vu==0,1 as seeds, we get the single soliton solution and the double soliton solution by method of variation of constants.In section four, we consider a Darboux transformation of 3×3 matrix spectral problem. By the spectral problem, We derive its Darboux transformation. Then using p=0, q=0 as seeds, we have the explicit solutions.