Analytic Study On Several Nonlinear Evolution Models In Fields Such As Optical Fibers And Condensates With Symbolic Computation

Propagation of solitons in the optical fibers, plasma and condensates can be described by the nonlinear evolution equations, such as nonlinear Schrodinger (NLS) equation. Solitons come from the balance between dis-persion and nonlinearity, with the energy and amplitudes both localized.In this paper, with the aid of analytic methods, we investigate solitons in some fields such as optical fibers, condensates based on several NLS-type equations. Certain of the results could be useful for understanding the generation mechanism and properties of solitons. The main research of the paper can be summarized as:(1) Based on an integrable higher-order NLS equation for a density-modulated quantum condensate, we investigate the propagation and in-teractions of dark solitons. According to the Ablowitz-Kaup-Newell-Segur(AKNS) system, the first three conservation laws and iteration relation are given. Through the Hirota method and symbolic computation, dark-soliton solutions are obtained by introducing an auxiliary function. We find that coefficients in the equation affect the soliton velocity. Via the graphical analysis, the propagation and collisions of dark solitons are dis-cussed.(2) Under investigation is a Heisenberg ferromagnetic spin chain with the bilinear and anisotropic interactions, which can be described by a(2+1)-dimensional NLS equation. By virtue of the bilinear method and symbolic computation, we construct multi-soliton solutions. Based on the analytic solutions, interactions between the two solitons are proved to be elastic through the asymptotic analysis. The one, two and three solitons are analyzed graphically and we find the amplitudes and widths of the two and three solitons keep invariant after each interaction.(3) Integrable properties and soliton solutions of the higher-order NLS equations in the birefringent or two-mode fiber are investigated.According to the AKNS system, 3x3 Lax pair for the NLS equations is derived. Based on Lax pair, we construct the conservation laws and Darboux transformation (DT). The one- and two-soliton solutions are obtained via DT and we graphically show the propagation of one soliton and bound-state two solitons.(4) Based on the coupled variable-coefficient NLS equations, we study the propagation and collisions of vector solitons in the inhomogeneous optical fiber. Introducing the auxiliary functions, we derive the bilinear forms and construct vector soliton solutions through the Hirota method and symbolic computation. We find the effect of variable coefficients on soliton structures. Bound-state solitons and interactions are analyzed graphically.(5) Two higher dimensional nonlinear evolution equations are inves-tigated. With the Bell polynomials and Hirota method, bilinear forms,Backlund transformations and soliton solutions for the (2+1)-dimensional Jaulent-Miodek equation are obtained. Propagation of the one soliton and elastic interactions between the two solitons are discussed graphically. For the (3+1)-dimensional Kadomtsev-Petviashvili equation, lump solutions are obtained based on the bilinear form. The conditions to guarantee analyticity and rational localisation of the lump solutions are presented.Plots with particular choices of the involved parameters are made to show the lump solutions and their energy distributions.(6) Based on the coupled Gross-Pitaevskii equations, solitons in the two-component Bose-Einstein condensate with an external potential are investigated. According to the non-isospectral AKNS system, Lax pair and conservation laws are constructed. With the aid of Hirota method and symbolic computation, soliton solutions are obtained. Influence of the harmonic and linear potentials on the bound states between the two solitons is analyzed.