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

This thesis mainly studied the Darboux transformation of a nonlinear wave equation related to a second-order spectral problem and its exact solutions.The main work is as follows:Firstly,in chapter 1,we briefly introduce the development history of soliton theory and the main solution methods of soliton equations,leading to the nonlinear wave equation which is related to the second-order spectral problems to be studied in this paper and its reduction integrable equation;Secondly,in chapter 2,we first find the Lax pair of this nonlinear wave equation,and then construct the gauge transformation matrix T with the expansion function form of the nth power of A.With the help of the gauge transformation between the two second-order spectral problems and rigorous proof of the space part and the time part,we can derive the Darboux transformation of the nonlinear wave equation.Using the reduction technique,we can obtain the Darboux transformation of the reduction nonlinear wave equation;Finally,in chapter 3,as an application,the Darboux transformation that we already got in chapter 2 can be used to find the isolated solution of the reduction nonlinear wave equation by selecting the appropriate" seed solution",and the graphics of the solution can be drawn using drawing software.