Analysis And Research Of Several Nonlinear Models Based On Symbolic Computation

Since the concept of solitons was created by Zabusky and Kruskal in 1965,con-siderable progress has been made in the study of various types of solitons as well as their applications in such fields as nonlinear optics,fluid dynamics,plasma physics,con-densed matter physics,solids and astrophysics.In contrast,to ordinary solitons which cnjoy a strong stability,rogue waves are the localized structures with the instability and unpredictability.Moreover,rogue waves have attracted much attention owing to their potential applications in various fields including the oceanography,nonlinear filer optics,condensed matter physics atmospheric dynamics,plasmas and finance.To get,the soliton and rogue wave solutions for the nonlinear partial differential equations,we attempt to implement some different methods which are analytical and numerical.The main contents are as follows:Chapter 1 is the introduction of the dissertation,where the origin,development and current situation of soliton and rogue wave are presented.Some analytical and numerical method are introduced to get the soliton and rogue wave solutions,such as bilinear method,modified bilinear method,mmerical simulation.The main work and the organization of this dissertation are also reported.In chapter 2,we ihvestigate a nonlinear fiber described by a(2+1)-dineensional complex Ginzburg-Landau equation with the chromatic dispersion,optical filtering,the nonlinear and linear gain.Backlund transformation in the bilinear form is construct-ed With the mdified bilinear method,analytic soliton solutions are obtained.For the soliton,the amplitude can decrease or increase when the absolute value of the nonlin-ear or linear gain is enlarged,and the width can be compressed or amplified when the absolute value of the chromatic dispersion or optical filtering is enhanced.We study the stability of the numerical solutions numerically by applying the increasing ampli-tude,embedding the white noise and adding the Gaussian pulse to the initial values based on the analytic solutions,which shows that the numerical solutions are stable,not influenced by the finite initial perturbations.In chapter 3,we consider the(2+1)-dimensional varia.ble-coefficient Nizhnik-Novikov*-Veselov equations in an inhomogeneous medium,which are the isotropic Lax integrable extension of the Korteweg-de Vries equation.Infinitely-many conservation laws are con-structed.N-soliton solutions in terms of the Wronskian are obtained via the bilinear Hirota method.Velocity and shape of the soliton can be influenced by those variable coefficients,while the amplitude of the soliton can not be affected.Collision between the two solitons is shown to be elastic.Breather wave solutions are constructed via the trilinear method.Such phenomena as the deceleration and compression of the breather waves are studied graphically.Rogue wave solutions are derived when the periods of the breather wave solutions go to the infinity.Separated and unite.d composite rogue waves are discussed graphically.In chapter 4,Under investigation in this paper is a(2+1)-dimensional variable-coefficient breaking soliton equation in fluids and plasmas.Bilinear forms,one-,two-and three-soliton solutions are derived via bilinear method.Velocity and shape of the soliton can be influenced by the dispersion coefficients ?(t)and ?(t),while the amplitude of the soliton can not be affected.Interaction between the two and among the three solitons are graphically discussed:the soliton has the phase shift without any influence on its shape and amplitude after the interaction,indicating that the interaction between the solitons is elastic.The stability of the soliton is studied numerically by considering truncation errors as small perturbations,which shows the soliton propagates steadily under the small perturbations.Chapter 5 is the conclusion,in which we sum up the contents and innovations of this dissertation and propose some further research work.