Analytic Study On Integrability And Soliton Solutions Of Some Nonlinear Evolution Equations

With the advancement of research on nonlinear phenomena, peo-ple have paid more attention to the dynamics of nonlinear phenomena for its potential applications in many fields, such as condensed matter physics, biology, fluid mechanism and astrophysics. Though the man-ifestation of nonlinear phenomena changes according to the different fields, common feature presented in the mathematical description is that they can be described by the corresponding nonlinear evolution equations (NLEEs). Through analyzing the NLEEs, we can under-stand the dynamic property of those nonlinear phenomena. Soliton theory, as an important aspect of nonlinear science, has attracted some interest for the role in explaining those nonlinear phenomena with good feature. In this paper, we will concentrate on the integra-bility of some coupled and higher-order NLEEs, and give their an-alytical soliton solutions by combining analytical method with sym-bolic computation. This dissertation mainly includes the following six parts: (1) To analyze the propagation of optical pulse in inhomoge-neous erbium-doped fiber, we consider a variable-coefficient nonlin-ear Schrodinger-Maxwell-Bloch equation and a variable-coefficient Hirota-Maxwell-Bloch equation. Under certain constraint and proper transformation, multi-soliton solutions of those two equations are ob-tained. Through the analysis on soliton propagation, the self-induced transparency effect caused by the doped erbium atoms is found to lead to the change of the soliton velocity and phase. Soliton interactions in different fibers are also investigated. In addition, we perform the linear stability analysis to study the modulational instability of the s-tationary solutions and give the condition, under which modulational instability occurs.(2) Painleve analysis is used to check the integrable property of some coupled and variable-coefficient NLEEs. An N-coupled non-linear Schrodinger equation is proved to be integrable in the sense of passing the Painleve test. In the process of performing Painleve anal-ysis on an inhomogeneous generalized fifth-order nonlinear Schrodinger equation, constraint on the coefficients, namely the integrable condi-tion of the equation, is given to make the equation pass the Painleve analysis.(3) Considering the simultaneous propagation of multiple pluses in nonlinear fibers, we analyze a general coupled nonlinear Schrodinger equation, which is a generalization of the Manakov equation and in-cludes the four-wave mixing terms. Soliton solutions are derived by the Hirota bilinear method. Through asymptotic analysis on the two- soliton solutions, we give the physical quantities of solitons before and after the collision and the condition, which decides whether elastic or inelastic collision occurs. Furthermore, a novel inelastic collision be-tween two solitons is found and discussed.(4) We also study an N-coupled nonlinear Schrodinger equation, which includes different self-phase modulation and cross-phase mod-ulation terms and can not reduce to the Manakov equation. The equation is verified to be integrable through constructing its Lax pair in the matrix form. By introducing an auxiliary function, the soliton solutions of the equation are obtained via Hirota bilinear method. With the help of asymptotic analysis, we classify the solitons and discuss the soliton interactions between the same and different types of solitons.(5) For the research on the dynamics of matter-wave soliton in Bose-Einstein condensates, a three-component Gross-Pitaevskii equa-tion is investigated. According to the differences of spin states, it is found that there are two types of solitons:ferromagnetic solitons in three components share the same pulse shape; polar solitons in three components have the one-or two-peak profiles, and the separated distance between two peaks is related to the polarization parameters. Soliton interactions between solitons in the same and different states are analyzed through asymptotic analysis.(6) We consider an inhomogeneous generalized nonlinear Schrodi-nger equation, which can describe the dynamics of a site-dependent Hisenberg ferromagnetic spin chain. Based on the Lax pair of the equation, an infinite number of conservation laws are constructed, which verify the integrable property of the equation in some mea-sure. Both Hirota bilinear method and Daxboux transformation are employed to derive the soliton solutions of the equation. The in-fluences of inhomogeneities and perturbation terms on the soliton propagation and interaction are discussed in both cases.In conclusion, we analytically study the integrable property and soliton solutions of some important NLEEs in such fields as optical communication, Bose-Einstein condensates and Hisenberg ferromag-netism via symbolic computation. The method utilized in this paper can be spreaded to the research on other coupled, high-order and variable-coefficient NLEEs. The result and analysis on the soliton solution obtained in this paper are expected to provide some help in the study of relevant fields.