The Exact Solution Of The Nonlinear Differential - Difference Equation

The nonlinear differential-difference equations not only have important applications in such sophisticated fields as engineering technology, automatic control^ space satellite, but also have become an indispensable mathematics tool in such natural science and domain of the social sciences as computer sciences x populmion dynamics and financial economics. In all of the theories above, how to get the solutions of the nonlinear differential-difference equations have more significant research values. But because the nonlinear differential-difference equations are semi-discrete with the spacial variables discrete while time is usually kept continuous, it is very difficult in solving the nonlinear differential-difference equations. So the study of solving the nonlinear differential-difference equations are valuable in theory and practice.In this paper, based on the mathematical mechanization, the exact solutions of the nonlinear differential-difference equations are studied in details. A new method of solving the equations- discrete mKdV auxiliary equation method is proposed. Applying the method, we solve the Self-dual network equation, coupled KdV-mKdV equation and discrete sine-Gordon equation and acquire some new exact solutions.The arrangement of this paper is as follows with four parts.In Chapter 1, the history and present situation of mathematical mechanization and soliton are mainly introduced initially.In Chapter 2, we research the history and present situation of the nonlinear differential-difference equations and discuss the theoretical and practical senses of the equation. Then, some approaches of solving the equations are summarized.In Chapter 3, we introduce the (^-expansion method and apply the method to solve the the Self-dual network equation and coupled KdV-mKdV equation.In Chapter 4, we study the discrete mKdV auxiliary equation method and use the method to solve the Self-dual network equation, coupled KdV-mKdV equation and discrete sine-Gordon equation.Finally, we summarize the new exact solutions, we can prove the effectiveness of the discrete mKdV auxiliary equation method and it can be applied to solving other nonlinear differential-difference equations.