quad physics, fluid dynamics, biology, chemistry, optics,quad physics, fluid dynamics, biology, chemistry, optics,

quad Theory of nonlinear dynamical systems comprises a broad range of numerical, geometrical, analytical and topological techniques for analyzing differential equations and iterated mappings. As a mathematical discipline, it should perhaps be viewed as a normal development within mathematics, rather than a scientific revolution or paradigm shift. The present structure or form of this theory have been determined through a successive analytical, experimental and numerical innovation in the last three decades. Aubin and Dahan Damlmedico cite{ADD02} study the socio–historical analysis on extra–mathematical influences and describe the confluence of ideas and traditions that occurred in western Europe and the US in the turbulent decade.The birth of nonlinear science lies in the fields of mathematics, physics, mechanics and computer science is in the 20th century since the mid 60th, especially in the late 70th emergence of mathematical sciences. However, crucial motivations and ideas have entered this area of mathematics from the applied sciences disciplines notably in classical mechanics, statistical physics, fluid dynamics, biology, chemistry, optics, atomic and molecular physics, environmental sciences, engineering sciences, in the context of both basic and applied investigations cite{Nic95, ADD02}. Partial differential equations also called mathematical physics equations are derived from the models of physics, mechanics and engineering etc which are described in a variety of phenomena. From the 20th century onwards a lot of new partial differential equations are derived, including the famous Maxwell system, Schr”{o}dinger equation, Einstein equation, Yang–Mills equation, and reaction-diffusion equation, etc. And these equations are all classical partial differential equations cite{TE83, Wei65}.Partial differential equations have become a useful tool for describing these natural phenomena of science and engineering models for instance, the heat flow and the wave propagation phenomenon, the dispersion of a chemically reactive material, most physical phenomena of fluid dynamics, quantum mechanics, electricity, plasma physics, propagation of shallow water waves and many other models are controlled within its domain of validity by partial differential equations  cite{Lam98, Wazw02}. Therefore, it becomes increasingly important to be familiar with all traditional and recently developed methods for solving partial differential equations and the implementation of these methods. It is enshrined in the nonlinear system commonality and some of the universality of theorems will become the refactoring mathematical sciences as the cornerstone to the natural sciences fields in some qualitative descriptive subjects of quantitative process.Its impact has a wide range of coverage in natural sciences and social sciences in almost all areas, nonlinear science is a challenge to the entire modern knowledge systems becomes as the new science cite{ADD02}. In nonlinear science, the study of nonlinear wave equation is very active; in particular, the soliton theory in various fields of natural science plays a very important role. On one hand, in nonlinear optics, flux quantum devices, biology, plasma, optical soliton communications and on the other side, the high–tech fields having a remarkable interest and application by physicists and mathematicians which greatly promotes the development of the traditional mathematical theory in a study of nonlinear wave equations.Inaddition, the nonlinear wave equations are important mathematical models for describing natural phenomena and one of the forefront topics in the studies of nonlinear mathematical physics, especially in the studies of soliton theory. The research on finding explicit and exact solutions of nonlinear wave equations and on analyzing their qualitative behavior can help to understand the motion laws of  systems under the nonlinear interactions, explain the corresponding natural phenomena reasonably, describe the essential properties of the nonlinear systems more deeply and promote greatly the development of engineering technology and related subjects.Various methods have been studied and developed to seek solutions of nonlinear partial differential equations and to study the characteristics and behavior of solution. Such as backscatter method, variables separation method, B”{a}cklund transformation method, Darboux transformation method, Hirota bilinear method, bifurcation method, etc.  There are enormous literatures on the study of nonlinear wave equations in which the above methods are applied. But, to our understanding the bifurcation method of dynamical systems has not been intensively applied in the study of the dynamical behavior, the parametric dependence of the travelling wave solutions, the smoothness of phase curves, the new areas of solutions types so–called compacton, peakon solutions and exact solution of those dynamical systems.