Soliton Solutions And Some Related Properties Of Nonlinear Evolution Equations Based On Symbolic Computation

With the development of modern science,more and more nonlinear evo-lution equations have been taken into consideration.For many years,people have complete research on linear systems,but in the real world,the nonlinear equations could describe many phenomenon in nature or even social science.Therefore,it becomes important to study the nonlinear theory.Nowadays,the chief applications of the nonlinear theories are:Fluid mechanics,nonlinear op-tical fibers,plasmas and lattice,etc.The soliton,known as an important branch of the nonlinear theory,has some priorities that the normal waves do not possess.It can be applied to clarify some phenomenon in consideration of energy and velocities during the propa-gation of the waves in the nonlinear systems.In recent years,in order to ob-tain the soliton solution of the nonlinear evolution equations,many approaches have been established such as Hirota direct method,Backlund transformation,Wroskian determinant,etc.This paper has introduced the research background and some of the ap-proaches briefly,then investigated the soliton solution of some nonlinear evo-lution equations.The main contexts are orgnized as follows:The first chapter introduces the history of the nonlinear evolution equa-tions,its current development situation and the common methods to solve the equations.The second chapter specificly introduces the Hirota direct method and the Backlund transformation,which are used in this paper.The third chap-ter introduces the generalized nonlinear Schrodinger-Maxwell-Bloch system,which describes the propagation of the optical solitons in an optical fiber doped with two-level resonant impurities like erbium with the fourth-order dispersion taken into account.Via the Hirota method,symbolic,computation.and intro-duction of the auxiliary function,Bilinear forms are derived.For the complex envelope of the field and the measure of the polarization for the resonant medi-um,bright solitons can be obtained,while dark ones have been deduced for the extant population inversion.Propagation of the one and two solitons is ana-lyzed.The fourth chapter consider a KdV-type extension,called the generalized(2+1)-dimensional variable-coefficient Nizhnik-Novikov-Veselov equations in an inhomogenous medium such as the incompressible fluid.Via the Hirota bilinear method and symbolic computation,we derive the bilinear forms,N-soliton solutions and Backlund transformations.Soliton evolution and interac-tions are graphically presented and analyzed:Interactions between the solitons are elastic.The soliton with larger amplitude moves faster than the smaller.When the first and high-order dispersion coefficients of those equations become t-dependent,velocities of the solitons vary with t and shapes of the solitons be-come curved.The fifth chapter focuses on a(3+1)-dimensional coupled nonlinear schrodinger system in an optical fiber.Via the Hirota direct method,dark one-and two-soliton solutions are obtained.Soliton properties and interaction are graphically discussed.The sixth chapter is the study of a(2+1)-dimensional coupled nonlinear Schrodinger equations in a graded-index waveguide.Mixed-type soliton solu-tions are obtained,soliton properties and interaction are discussed.The final chapter is a summary of the main points of the paper.