The Study Of Analytic Solutions For Nonlinear Equations And Their Physical Applications Based On Symbolic Computation

Nonlinear phenomena exist widely among the various areas of mathematics and physics, thus, the study on them is one of the hotspots of the academia. In particular, the soliton theory also attract a lot of attention from scholars. Soliton theory research scope has been expanded to fluid mechanics, nonlinear optics and other fields, and guided the development of engineering applications. Therefore, it is still a research topic with great value that how to solve the nonlinear evolution equations to get soliton solution.In this paper, based on Hirota bilinear method and computer symbolic computation method, we will investigate the spatial optical solitons in photorefractive polymer, the two-dimensional matter-wave solitons in Bose-Einstein condensate, and the spatiotemporal soliton in inhomogeneous Kerr media. Through derivation of the soliton solutions for the three types of nonlinear equations, and then through asymptotic analysis and graphic analysis, we will study the impact of the model parameters on the solitons.The chapters and main contents of this paper are organized as follows:The first chapter is the introduction of the background and status. Two methods of nonlinear evolution equations are given, the inverse scattering transform method and Hirota bilinear method.The second chapter will be the study on coupled nonlinear Schrodinger equations. The bilinear forms, bright and dark soliton solutions of the equations are obtained. Based on the choice of photorefractive polymer parameter and incident-optical-beam parameter, analysis on the soliton propagation is carried on. The analysis on elastic collision and inelastic collision between two solitons is given. Finally, we do the solution stability analysis.The third chapter will be the study on the two-dimensional Gross-Pitaevskii equation with variable coefficients. We derive its one-, two-and three-soliton solutions and analyse the time-dependent harmonic trapping potential parameters on one soliton’s evolution. Collision properties of the two and three2D matter-wave solitons are revealed graphically in different cases.The fourth chapter will be the study on the two-dimensional nonlinear Schrodinger equation with variable coefficients. Via the Hirota method and symbolic computation, analytic bright one-and two-soliton solutions for such an equation are obtained. Based on the one-soliton solutions, soliton dynamics with different choices of the group velocity dispersion coefficient and Kerr effect parameter is discussed. Through the graphic and asymptotic analysis on the two-soliton solutions, the interactions between two solitons are illustrated.In the final chapter, we will give a summary of this paper to point out the significance and innovations of this paper.