| It is usually very difficult to get explicit solutions for nonlinear partial differential equations. There are several systematic approaches to obtain explicit solutions of the soliton equations such as the inverse scattering transformation, the Hirota technique, the Darboux transformation, the algebra-geometric method, and so on. Among the various approaches, Darboux transformation has been proved to be a powerful tool in finding non-trivial expilcit solutions of soliton equations from a trivial seed. In this paper, we derive a hierarchy of new nonlinear evolution equations associated with a 3 x 3 matrix spectral problem. With the help of the trace identity, it is shown that the hierarchy of nonlinear evolution equations has the generalized Hamiltonian structures. The first nontrivial equation in the hierarchy is whose Lax pair is the spectral problem and the auxiliary problem where u, w, v are the functions of x, t,λa constant. We prove that the equation, Tx+TU= UT, holds, where U and U have the same form except changing u, v, w into u,v,w. Based on the fact we obtain a Darboux matrix with multi-parameters: and a Darboux transformation: where Tij(0),(i,j=1,2,3) are the functions of x and t.As an application of the Darboux transformation,we get nontrivial explicit solutions of this nonlinear evolution equation from a trivial see u=v=w=0. |
