Darboux Transformation Of Nonlinear Evolution Equation Associated With A Third Order Spectral Problem And Its Exact Solutions

This thesis mainly studied the Darboux transformation of a nonlinear evolution e-quation associated with a third order spectral problem and its exact solutions.First,the Lax pairs of the nonlinear equation are given,and the soliton equation can be obtained according to the compatibility.Secondly,based on the Lax pairs of the nonlinear equation,the gauge transformation of the spectral problems is constructed,and the solving problem of the gauge transformation matrix is transformed into the solving problem of a linear system.Then the Darboux transformation of the nonlinear evolution equation is given,and it is proved completely in theory.Finally,the exact solution of the nonlinear evolution equation is obtained from the "seed solution" by using the derived Darboux transform.And the graph of the solution is drawn.