Symbolic Computation On The Exact Solutions And Soliton Dynamics Of The Nonlinear Evolution Equations

During more than a hundred years since soliton has been found, the soliton theory as an important branch of the Nonlinear Science has a great development. The soli-ton solutions have been investigated and widely applied in many fields including fluid mechanics, plasma physics, optical communications and other fields of natural science. On the standpoint of mathematics, the energy of soliton is limited and distributes in the finite space, and soliton collisions belong to elasticity collisions after which the soli-ton can recover the wave form and propagation velocity. However, the soliton with metamorphic wave form in the propagation process is still called soliton in physics as researchers focus on the soliton quantum state nor the changing of the wave form in physical investigation. Soliton has been well studied and comprehensive applied based on the favourable physical properties. Seeking for the exacts solutions of the nonlinear evolution equations(NLEEs) has become the key point of the investigation on nonlinear science, and the difficult one meanwhile. Hitherto, there are many methods to obtain the exact solutions for the NLEEs such as the homogeneous balance method, Darboux transformation method, inverse scattering method, Backlund transformation method, Exp-function method, Hirota method and so on.Based on the theory of the NLEEs, with symbolic computation this paper studies several NLEEs applied in fluid mechanics, nucleic fission and optical communications via Exp-function method and Hirota method and reveals the soliton solutions and analyse the properties of the soliton interactions.The structure of the present paper is organized as follow:In chapter one, we first introduce the history and development of the soliton the-ory and the current situation, and then we expound Exp-function method and Hirota method for solving the NLEEs. At last, we describe the overall works and arrangements.In chapter two, we study the general MKdV equation via Exp-function method and symbolic computation and receive the analyzing solutions. Through changing the parameters new soliton solutions and periodic solutions have been derived, especially 1-and 2-dark soliton solutions. And then analyzing the soliton propagation characters by numerical simulation.In chapter three, we research the Sharma-Tasso-Olver(STO) equation via Exp-function method and symbolic computation and receive the analyzing solutions. By numerical simulation the propagation mechanisms of new 1- and 2-oliton solutions have been revealed. However, based on a great number of symbolic computation existing in the process of seeking for N-oliton (N≥2) via Exp-function method, which reduces the efficiency of solving the NLEEs, thus we have derived the bilinear form via Hirota method and discovered the information can be referenced.In chapter four, we study the coupled nonlinear schrodinger (CNLS) equations via Hirota method. Through variable transformation the bilinear form has been derived. Then N-dark soliton solutions(N=1,2,3), breatheres and the collisions between the soliton and the breather have been discovered certainly by symbolic computation. And we analyze the propagation characters of the solitons and the influence of parameters by simulating the solutions.In chapter five, we study a general coupled Hirota (GCH) equations via Hirota method. Through variable transformation the bilinear form has been derived. Then N-soliton solutions(N=1,2,3), breatheres and the collisions between the soliton and the breather have been discovered certainly by symbolic computation. And we ana-lyze the propagation characters of the solitons and the influence of parameters on the propagation velocity and the wave form by simulating.In chapter six, we conclude innovations as follow:1. Through Exp-function method we have obtained the exact solutions for the general MKdV equation, especially the N-dark soliton solutions(N=1,2) and periodic solutions; 2. Through Exp-function method we have obtained the soliton solutions for the STO equation, especially the dark soliton solutions and new 2-soliton solutions, and then received the bilinear form via Hirota method; 3. Based Hirota method we have obtained the N-soliton solutions and breathers for the CNLS equation and clarified the soliton interaction by numerical simulation; 4. Based Hirota method we have achieved the N-soliton solutions(N=2,3) and breathers for the GCH equation and clarified the soliton interaction by numerical simulation.