Darboux Transformation And Exact Solutions Of Nonlinear Equation Associated With Third Order Semi-discrete Spectral Problems

This thesis mainly studied the Darboux transformation and its exact solution of a nonlinear evolution equation associated with a semi-discrete 3 × 3 matrix spectral problem.Firstly,according to the spectral problem and its auxiliary spectral problem,a nonlinear evolution equation is derived by using the semi-discrete zero curvature equation.Then,starting from the Lax pairs of this equation,the gauge transformation Tn of the spectral problem is constructed,so that the 1-fold and 2-fold Darboux transformations of the nonlinear equation are obtained and the related proofs are given.Finally,based on the suitably choosen seed solution,new exact solution of the nonlinear evolution equation associated with the semi-discrete 3 × 3 matrix spectral problem is given by means of the Darboux transformation.