Bifurcation Analysis And Some Exact Solutions Of Some Nonlinear Evolution Equations

With the development of science, the intricate phenomena in nature have stimulated people to explore their essence, which makes the nonlinear science produce and flourish. Compared with the linear system,the nonlinear models can describe the natural phenomenon better and more precisely, and are closer to the essence of the phenomenons.Nonlinear partial differential equations play an important role in the field of nonlinear science, and the soliton theory is an important direction for the study of nonlinear partial differential equations. Because in-depth researches on natural science and engineering applications are often incorporated into the study of nonlinear partial differential equations, and nonlinear equations are extremely complex,it becomes difficult to find a very effective method to solve exact solutions,so how to obtain the exact solutions of nonlinear evolution equation is the primary task of studying nonlinear problems.As early in 1834, solitary wave phenomenon discovered by the British scientist Russell As early in 1834 has again aroused great concerns in the past 20 years, scientists's interests in this phenomenon are growing. This is because on the one hand,solitons have many properties of particles and waves, and they have universality in nature. Soliton theory has also successfully explained many phenomena that can not be solved by classical theory in physics for a long time. On the other hand,With the deep study of the problem of soliton physics, the mathematical theory of soliton has come into being, and has formed a relatively perfect theoretical system.Poincare pioneered the use of qualitative ideas and geometric methods to explore the properties of solutions from the differential equations themselves. When the differential equation is linear, the analytic solution can be obtained by some common methods such as Laplace transform or series solution. That is, the solution of the differential equations studied can be expressed in a mathematical formula.When the differential equation is nonlinear, the analytic solution can not be obtained under normal circumstances. At this time, the problems can only be studied by some auxiliary means. One approach is to use numerical approximation. With the popularization of computer and the support of some powerful calculation software, the numerical solutions of nonlinear differential equation are easy to be obtained, and its nature is also easy to see. But in many application problems, such as models discribed by the nonlinear differential equation in biology, chemistry,physics and other natural science, when the analytic solutions of the model are not the case, our interest in the problem is changed to the discussions on the whole properties of system which are the so-called qualitative theory, such as whether the system has periodic solutions,The number of solutions and so on. So the qualitative theory of dynamical system is also important and necessary. The research of this paper mainly includes:(1)Firstly, the dynamical behavior of a MS equation is studied by means of the theory of geometric bifurcation and the dynamical system qualitative theory, and some new exact solutions are obtained by applying the new auxiliary function method, Jacobi elliptic function method and the simplest equation method. By a traveling wave transformation, we convert the partial differential equation into the ordinary differential equation, then we draw the trajectory diagram, determine the type of solution, and get the exact solutions of solitary wave solutions and periodic solutions of elliptic function.(2)Secondly, the dynamic behavior of a Zhiber Shabat equation is studied by means of the theory of geometric bifurcation and dynamical system. Some new exact solutions are obtained by applying the ITEM(a new method for solving nonlinear equations) metheod and generalized sine-cosine method. By traveling wave transformation, we convert the partial differential equation into an ordinary differential equation, then we draw the trajectory diagram, determine the type of solution, and get the exact solutions of solitary wave solutions and periodic solutions of elliptic function.(3)Finally, we study a new phenomenon in the field of nonlinear-rogue waves. We take the second kind of KP equation as an example. By using bilinear transformation, the homoclinic (heteroclinic) breather wave solutions of the KP equation are obtained by using the homoclinic breather limit method, and then we get the rogue wave solutions by making the period of periodic wave solutions going to infinite.In this paper, some nonlinear evolution equations with important researching values in optical communication and hydrodynamics are analyzed interdisciplinarily based on combination of the several construction methods of exact solutions of nonlinear equations and computer symbol computation,including their dynamical system analysis,analysis of the nature of the solutions and the corresponding between solutions and phase portrait. The research methods used in this paper can also be applied to other physical or engineering researches on a number of complex nonlinear models. At the same time, for the results obtainedin this paper the author hopes to provide some theoretical help for the research in this field.