Analytic Study On Nonlinear Evolution Equations Based On Bilinear Method And Dynamical System

Nonlinear science, a subject of nature science field, is a new interdisci-plinary subject to study the generality of nonlinear problems. The nonlinear evolution equations which we research mainly generate from such fields as flu-id mechanism, plasma physics, nonlinear optics, condensed matter physics, and biology. Nowadays, many methods have been applied to solve different kinds of nonlinear evolution equations, for example, the inverse scattering method, Backlund method, Darboux transformation, Hirota bilinear method, Painleve analysis, bifurcation theory and homoclinic breathers limit method. In this pa-per, based on the theory of nonlinear evolution equations, using Hirota bilinear method, Painleve analysis, bifurcation theory and homoclinic breathers limit method, and through symbolic computation, we will study several nonlinear evolution equations, obtain the analytical solutions and further research the ba-sic properties of the solutions. This dissertation mainly includes the following three parts:(1) We discuss a nonlinear Schrodinger equation with Kerr law nonlinear-ity which describes the transmission of lightwave in a nonlinear optical fiber by using bifurcation theory and qualitative theory of dynamical systems. By traveling wave transformation, partial differential equations will be converted to ordinary differential equations. Then we draw phase portraits of the ordinary system and discriminate the type of the solutions. Finally, we obtain solitary wave solutions and periodic solutions of elliptic functions.(2) Secondly, we study the coupled modified Korteweg-de Vries equation- s, which describe the interaction of waves in the transmission process. We deal with the coupled modified Korteweg-de Vries equations by two kinds of traveling wave transformation and get the ordinary differential system. Under different parameters, we get six groups of phase portraits. According to those phase portraits, we get solitary wave solutions and periodic solutions of elliptic function. In addition, by using extended elliptic function method, we obtain new periodic solutions of elliptic functions.(3) Finally, we discuss a special water wave—rogue waves. We will dis-cuss a (2+1) dimensional Kadomtsev-Petviashvili equation and a (1+1) dimen-sional Symmetric Regularized-Long-Wave Equation. Based on Hirota bilinear method, symbolic computation, and the homoclinic limit breathers method, we obtain the breather solution and then get the rogue wave solution.In conclusion, we analytically study the solutions of some important non-linear evolution equations in such fields as optical communication and plasma physics via symbolic computation. Research methods used in this article can also be applied to study other couple equations and higher order nonlinear mod-el. The results and analysis on the soliton solutions obtained in this paper are expected to provide some help in the study of relevant fields.