Analytic Studies On The Soliton Problems In Some Nonlinear Systems

Soliton, as a kind of solutions for the nonlinear system, can be used to describe certain nonlinear phenomena in such fields as Bose-Einstein condensates, optical fiber communication, Heisenberg ferro-magnetic, plasmas and fluid dynamics. Studies on soliton in different systems need the help of the study on the nonlinear evolution equa-tions. Nowadays, analytic methods have been reported to derive the soliton solutions for the nonlinear evolution equations, for example, the Lax pair, Darboux transformation, Hirota method, Bell polyno-mial approach, Backlund transformation, and so on. Based on these methods, we will aim to study the soliton solutions and integrabilities for some nonlinear evolution equations. Moreover, the dynamics of the soliton propagation and interactions will be discussed.The main research work of this paper is given as:(1) Under investigation are the coupled Gross-Pitaevskii equa-tions, which describe the dynamics of two-component Bose-Einstein condensates. Infinitely-many conservation laws are obtained based on the Lax pair. Via the Bell-polynomial approach and Hirota method, Bell-polynomial-typed transformation and bilinear-typed Backlund transformation are derived. Single soliton solutions are obtained via the iteration process from the seed solutions. One-and two-soliton solutions are expressed explicitly. Head-on and overtaking elastic interactions are shown and analyzed from graphics. Inelastic interac-tions between two soliton-like envelopes are presented as well.(2) Soliton propagation and interaction in optical fiber commu-nications are investigated. The research objects are:coupled higher-order nonlinear Schrodinger equations with variable coefficients, an inhomogeneous higher-order nonlinear Schrodinger equation and N-coupled generalized nonlinear Schrodinger equations with cubic-quintic nonlinearity. (a) Infinitely-many conservation laws of the coupled higher-order nonlinear Schrodinger equations with variable coefficients are obtained through the Lax pair. With different coefficients, bell-shaped, periodic-changing, quadratic-varying, exponential-decreasing and exponential-increasing soliton profiles are seen, to describe the propagation and interactions of the femtosecond soliton pulses. Via the exchange formula, bilinear Backlund transformation are obtained. (b) By virtue of gauge transformations and auxiliary function, we get the bilinear forms for an inhomogeneous higher-order nonlinear Schrodinger equation, the nondegenerate-soliton solutions are also derived. The single-hump, double-hump and flat-top profiles are dis-played through figures. Elastic interactions between a single-hump soliton and a double-hump soliton, between the two double-hump soli-tons are analyzed.(c) Via the modified Darboux transformation, the first- and second-order vector rogue waves for the N-coupled general-ized nonlinear Schrodinger equations with cubic-quintic nonlinearity are analytically presented. Two different patterns of the second-order vector rogue waves are analyzed graphically. Finally, the modulation instability of the plane-wave solutions is given.(3) Under investigation is an extended higher-order nonlinear Schrodinger equation, which describes the spin dynamics of a weak anisotropic Heisenberg ferromagnetic. Based on the Lax pair, infinitely-many conservation laws are obtained. With the aid of an auxiliary function, multi-soliton solutions are presented explicitly. When the adjacent solitons propagate with the same velocity, bound states of the bright solitons occur. The velocity, interaction periods and sepa-ration of the bound states are expressed. In addition, the condition for the modulation instability is given.(4) An inhomogeneous nonlinear Schrodinger equation in an inho-mogeneous plasma is investigated. Backlund transformation and N-soliton solutions are obtained. Periodically attractive and repulsive interactions between the two and three solitons are shown. Influence of the linear density coefficient and damping coefficient on the soliton envelopes is also discussed.(5) A discrete integrable Ablowitz-Ladik equation is investigated. We find that the discreteness effects make the influence on the soliton amplitude and velocity, furthermore, affect the soliton propagation and interaction behaviors.(6) Soliton fusion and fission in a generalized variable-coefficient fifth-order Korteweg-de Vries equation are investigated. Via the Bell-polynomial approach, Backlund transformation, Lax pair and infinitely-many conservation laws are derived. The conditions and types of the soliton fusion and fission, and influence of the variable coefficients from the equation are analyzed.(7) Introducing the auxiliary variable in Bell-polynomial approach and Hirota method respectively, we get the bilinear forms for the Zhiber-Shabat equation and the coupled Kadomtsev-Petviashvili-Schrodinger equation successfully, (a) For the Zhiber-Shabat equa-tion, the bell-shaped soliton, upside-down bell-shaped soliton and breather-like solutions are obtained. Figures are plotted to illustrate the elastic interactions between two upside-down bell-shaped soli-tons, (b) For the coupled Kadomtsev-Petviashvili-Schrodinger equa-tion, the integrability is analyzed by the Painleve test. Based on the N-soliton solutions, the interaction processes of the bright and dark solitons are illustrated respectively.