Solutions And Dynamic Analysis Of A Class Of Equations Related To The General Kaup-Newell Spectral Problem

Since the development of the soliton theory,many different nonlinear evolution equations with internal relations correspond to the same spectral problem.Therefore,whether it is through the Hirota method or by using the Wronskian technique combined with the knowledge of algebra,it is very important to solve the multiple forms of accurate solutions and the diversity of solutions for a class of soliton equations with the same spectral problem.The main research contents in this dissertation consist of: the N-soliton solution of the coupled Gerdjikov-Ivanov equation are obtained through the Hirota method.The(general-)double Wronskian solution of the coupled Gerdjikov-Ivanov equation is got by means of the Wronskian technique and more exact solutions are found by determining the different coefficient matrix of the conditional equations satisfied by the Wronskian elements.Firstly,based on the compatibility condition of the general Kaup-Newell spectral problem and the time evolution equation,the bilinear form of the coupled Gerdjikov-Ivanov equation is derived.By introducing the perturbation expansion,the Hirota-form N-soliton solution of the equation is obtained,and the dynamic characteristic of the single-soliton solution and the double-soliton solution are given.Then the N-soliton solution of the Gerdjikov-Ivanov equation is obtained by reduction.In order to find more exact solutions of the coupled Gerdjikov-Ivanov equation,the fourth chapter obtains the N-soliton solution expressed by the double Wronskian through using the Wronskian technique from the bilinear form of the equation,and the consistency is discussed between the Hirota-form solution and the solution expressed by the double Wronskian.Similarly,the double Wronskian solution of the Gerdjikov-Ivanov equation is obtained by reduction.Furthermore,the coefficients of the conditional equations satisfied by the elements in the double Wronskian solution are extended to the matrix form,and the general double Wronskian solution is derived.Finally,based on the theory of the general double Wronskian solution,the coefficient matrix of the conditional equations is taken as the Jordan matrix,and the rational solutions and Matveev solutions of the coupled Gerdjikov-Ivanov equation are obtained.More generally,if the coefficient matrix is taken as the quasi-Jordan matrix,and the Complexiton solutions of the equation are derived.Finally,the interaction solutions of the coupled Gerdjikov-Ivanov equation are obtained by constructing the mixed matrix composed of the Jordan matrix and quasi-Jordan matrix.