Study Of Some Nonlinear Evolution Equations Based On Symbolic Computation

Nonlinear phenomena, which exist widely among the various areas of mathematic and physics, have been one of the hotspots in academia. In par-ticular, the soliton theory, as an important aspect of nonlinear science, has at-tracted some interest for the role in explaining those nonlinear phenomena in such fields as condensed matter physics, biological sciences, fluid mechanics and astrophysics. This dissertation mainly includes the following five parts:(1) The first chapter is the background and status. The methods of nonlin-ear evolution equations are given, including symbolic computation and Hirota bilinear method.(2) The second chapter will be the study of propagation properties and in-teractions of the solitons formed by the incoherently interacting optical beams in the bulk Kerr and saturable media in nonlinear optical fibers, which can be governed by a (2+1)-dimensional N-coupled nonlinear Schrodinger system. Analytic mixed-type vector soliton solutions for such a system are derived. Then, we graphically illustrate the propagation properties and interactions of the mixed-type vector solitons. Through the analysis on the vector solitons, we find that the soliton amplitude and width are found to depend on the index of refraction. Inelastic and elastic overtaking interactions between the bright two solitons, and elastic oblique interaction between the dark two solitons, are also illustrated.(3) The third chapter will be the study on the high-dimensional solitons in the optical waveguides and Bose-Einstein condensates (BECs) governed by a (3+1)-dimensional Gross-Pitaevskii system. With the symbolic computation and Hirota method, analytic bright one-and two-soliton solutions under some certain conditions are derived. Soliton-amplitude/width amplification and influ-ence of the time-modulated dispersion on the bright-soliton shape are studied via the graphic analysis.(4) The fourth chapter will investigate a (3+1)-dimensional variable-coefficient Kadomtsev-Petviashvili equation, which describes the propagation of surface and internal water waves. By virtue of the binary Bell polynomi-als and symbolic computation, the bilinear forms, soliton solutions, Backlund transformations and Lax pair are obtained. We will also study the effect on solitonic structure from the variable coefficients and elastic interactions be-tween/among two and three solitons.(5) In fifth chapter, we mainly study the soliton propagation properties and soliton interactions for a (2+1)-dimensional nonlinear Schrodinger system, which can describe the dynamics of a nonlinear photonic quasicrystal or vor-tex Airy beam in the nonlinear optics. Analytic bright N-soliton and dark two-soliton solutions are firstly derived. Graphic description of the soliton properties and interactions in a nonlinear photonic quasicrystal or vortex Airy beam is illustrated. Then, the effects of the optical wavenumber/linear oppo-site wavenumber and nonlinear coefficient on the soliton amplitude and width are illustrated. Overtaking/periodic interactions between the bright two solitons and overtaking interaction between the dark two solitons are also illustrated.In conclusion, we will give a summary of this paper to point out the signif-icance and innovations of this paper.