The Exact Solution Of Nonlinear Wave Solution And The Study Of Its Dynamics Properties

With the development of science and technology, many nonlinear problemsarouse much concern. These matters are usually ascribed to nonlinear partialdifferential equations. Because of the complexity of nonlinear partial differentialequations, a lot of troubles are brought to solve the equation. In recent years, lots ofeffective ways to solve partial differential equations are given, and many nonlinearproblems are solved. But solving nonlinear equations (especially nonlinear partialdifferential equations) is very difficult, and there is no unified method, the methodbefore can only be applied to a certain or some equation. So, continuing to look forsome effective method of solving nonlinear equations is an important and meaningfulwork.In this paper, we only pay attention to the travelling wave solutions of the partialdifferential equation. Although people put forward a lot of method of solvingtravelling wave solutions, whether the nature of the travelling wave solutions for thesame nonlinear partial differential equation are different from each other and whetherthey reflects the completely different physical phenomena? There is little answer tothis question.Using the method of dynamical systems, the exact travelling wave solutions forthe (3+1) dimension Jimbo-Miwa and (2+1) dimension Gardner equation are studied.The sufficient conditions to guarantee the existence of smooth solutions are achievedby studying the phase diagrams in different parameter regions, and the exact explicitparametric representations of some solitary wave solutions and periodic wavesolutions with different parametric are given.The results show that the nature of the travelling wave solutions for a specificpartial differential equation is limited, thus, there is only a few classes travelling wave solutions for a partial differential equation. Therefore, we can guess that there must bean overlap or cover phenomenon whit many method, and some method can not findnew solutions, even some solution may be equivalent.