The Integrable Discretizations Of Two Soliton Equations

In this paper, two soliton equations are integrable dispersed by using Hirotamethod. In Chapter1, firstly, the emergence and development of the soliton and themethods to solve the nonlinear evolution equations are simply introduced, then anoverview to the methods for solving nonlinear differential–difference and nonlineardifference–difference equations are given, lastly, the main contents of this paper areintroduced. Chapter2introduces some basic knowledge and concepts, as well as theimportant formulas and properties needed in this paper. Chapter3elaborates theprocess of integrable discretization to mKdV equation by Hirota method, i.e. firstly anappropriate transformation is used to transform the mKdV equation into thecontinuous bilinear derivative equations, and then the bilinear derivative equations areintegrable dispersed by a hyperbolic operator. Finally, the soliton solutions are gottenby Hirota perturbation method, and then the integrable property is proved. Similar tochapter3, in chapter4, the second order AKNS equation is integrable dispersed byHirota method. Firstly, a family of new solutions of the differential–differencebilinear derivative equation of the second order AKNS equations are given by Hirotamethod; then, the difference–difference discrete bilinear derivative equations areobtained by dispersing the variable of time in the differential–difference bilinearderivative equation, and the exact solutions of the equations are given by Hirotamethod; lastly, the difference–difference AKNS equations are obtained by anappropriate transformation.