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

This paper mainly applies Darboux transformation to study a nonlinear evolution equation associated with the second-order spectral problem.By introducing the gauge transformation of the second-order spectral problem to derive the Darboux transformation of the nonlinear evolution equation,thereby giving a system algorithm for the exact solution of nonlinear evolution equation associated with the second-order spectral problem.As the application of Dardoux transformation,the nonlinear evolution equation is accurately solved,the main work is as follows:Firstly,the Lax pair of the nonlinear evolution equation is introduced,and then a gauge transformation about the N-th power of the spectral parameter is constructed.With the help of the gauge transformation between the spectral problem and the auxiliary spectral problem,the derived Darboux transformation of the nonlinear evolution equation is strictly proved.In addition,using the reduction method,the Darboux transformation of the reduced equation can be obtained.Finally,a specific application of Darboux transformation is given,by selecting appropriate parameters to obtain the corresponding "seed solution",the soliton solution of the reduced equation can be obtained by using the obtained Darboux transformation.The solution is verified and the graph of the solution is drawn by maple and mathematica software to reflect its dynamic characteristics.