Two Nonlinear Equations, Darboux Transformation And Soliton Solutions

The soliton equation is one of the most prominent subject in the fields of nonliear science. In this paper, we consider two important nonlinear equations. It is well known that there are several systematic approaches to obtain solutions of nonlinear equations. Darboux transformation(DT) has been proved to be one of the most natural and beautiful method to get explicit solutions of some equations from a trivial seed.In section one, we introduce the initial Darboux transformation method and Darboux matrix method. Based on these methods, in the following paper, Darboux transformations of two nonlinear equations are constructed.In section two, consider the Darboux transformation of Boussinesq-Burgers equationWith the help of B_N-1=1/2(u_x+v),C_N-1=1/2, we give a complete proof of Darboux matrix( α, A_k, B_k, C_k and D_k (0 ≤ k ≤ N -1) are the functions of x and t ).From a trivial seed u = 0,v = -1 , we use the Darboux transformation to generate new multi-soliton solutions of Boussinesq-Burgers soliton equation, then discuss the first two cases N =1 and N = 2. When N = 2, suitably choosing parameters, we plot beautiful three-soliton collision figures, especially soliton solution v|-[2] is the three-soliton which every soliton has two peaks. It is a new type for soliton figures associated with 2×2 matrix spectral problem.In section three, consider a Darboux transformation of the coupled dispersionless equationthe Darboux transformation is about the negative exponent of A. when construct and proof the Darboux transformation, we need much strategic. At last, choose g=l,r = 0asa seed, by use of the Darboux transformation, we get single soliton solutions, then together with the theory of Darboux matrix, we get double soliton solutions of the coupled dispersionless equation.