Darboux Transformation Of The Generalized Td Hierachy And Some Nonlinear Evolution Equations And Its Explicit Solutions

The soliton equation is one of the most prominent subject in the fields of nonliear science. In this paper, we consider the generalized TD hierarchies and some nonlinear evolution 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 the generalized TD hierarchies and some nonlinear evolution equations are constructed.In section two, consider the generalized TD hierarchiesUnder the proper transformation, we get some (2 +1)-dimensional or (1 + 1)-dimensional nonlinear evolution equations, and get the relation of the solution of the generalized TD hierarchies and some nonlinear evolution equations.In section three, with the help of we give a Darboux matrix of the generalized TD hierarchies(γ, A_k, B_k, C_k and D_k (0≤k≤N - 1) are the functions of x, y and t), then we get the Darboux transformation of the generalized TD hierarchies and some nonlinear equations.In section four, from a trivial seed u, v are constants, and v≠0, we use the Darboux transformation to generate explicit solutions of the generalized TD hierarchies and some (2 + 1)-dimensional or (1 + 1)-dimensional nonlinear equations, especially, we get new solutions of Kadomtsev-Petviashvili (KP) equation, then discuss the first two cases N = 1 and N = 2. When N = 1, suitably choosing parameters, we get single soliton solutions and plot the figures. When N = 2, suitably choosing parameters, we plot beautiful three-soliton collision figures of the generalized TD hierarchies.