Some New Integrable Nonlinear Evolution Equations And Darboux Transformation

In this paper, we derive two classes of new nonlinear evolution equations associated with a 4×4 matrix spectral problem. With the help of the trace identity, it is shown that the two classes of nonlinear evolution equations have the generalized Hamiltonian forms. A Miura transformation for the first nonlinear evolution equation in the first class and its modified equation are found. Using the gauge transformation of the spectial problem, a Darboux transformation of the first nontrivial nonlinear evolution equation is obtained. As an application of the Darboux transformation, some explicit solutions of the first nontrivial nonlinear evolution equation are got.