Integrable Analysis And Soliton Interaction Study On Nonlinear Systems

Soliton is an important branch of nonlinear science. It is widely used in mathematics and physics, and is also the theoretical foundation in financial field for studying the characteristics of market evolution. Therefore, the study of soliton theory is of great significance.This dissertation analytically investigates the integrability and soliton interaction mechanism in nonlinear systems. Through the analysis on some coupled and variable-coefficient nonlinear evolution equations (NLEEs) or nonlinear hierarchies, a series of results are derived, such as the integrable properties, analytical soliton solutions, Hamil-tonian structure and Liouville integrability. This dissertation mainly includes the fol-lowing six parts:(1) The construction of Darboux transformation (DT) and its applications in the cou-pled NLEEs.(a) The construction of DT for isospectral and nonisospectral integrable systems is investigated, taking the Hirota-Maxwell-Bloch (H-MB) equation, generalized inhomogeneous H-MB equation and inhomogeneous coupled NLS equation for example;(b) Based on the obtained one-and two-soliton solutions of the H-MB equation via the DT, the production mechanism, propagation characteristics and soliton interactions are investigated. The analysis is made to probe the influence of inhomogeneities in the generalized inhomogeneous H-MB equation on the soliton propagation and interac-tions via some figures. With the corresponding parameters of group velocity dispersion, self-phase modulation, cross-phase modulation and gain/loss under control, the soliton interaction mechanism and its potential applications of the inhomogeneous coupled NLS equations are discussed;(c) Via the Painleve analysis, the integrable conditions for the generalized inhomogeneous H-MB equation are obtained. Based on the Ablowitz-Kaup-Newell-Segur (AKNS) system, the Lax pairs associated with the H-MB equations and generalized inhomogeneous H-MB equations are constructed. With the2×2isospectral AKNS spectral problem generalized to the3×3nonisospectral case, the Lax pair for the inhomogeneous coupled NLS equations is derived. (2) The construction of N-fold DT and asymptotic analysis of the generalized in-homogeneous H-MB equation.(a) The N-fold DT for the generalized inhomogeneous H-MB equation is derived, as well as one-, two-and three-soliton solutions, which can be compiled into the determinant forms;(b) Soliton propagation characteristics and in-teractions with different parameters are discussed, and energy redistribution between the two solitons in their interactions is observed;(c) The asymptotic analysis is used to discuss some physical quantities before and after soliton interactions, such as the energy, amplitude, width, velocity and initial phase;(d) The first three conservation laws of the equation are obtained.(3) Nonlinear hierarchies and infinite conservation laws.(a) By taking KdV and AKNS hierarchies for example, the processes of constructing the nonlinear hierarchies with the Lax pair in operator or matrix forms are introduced;(b) By taking KdV and AKNS systems for example, the construction processes of the infinite conservation laws with the Lax pair in operator or matrix forms are investigated, and the infinite conservation laws for H-MB equation are also obtained;(c) Based on the generalized discrete spectral problem, the nonlinear hierarchy and infinite conservation laws of the discrete system are derived.(4) Hamiltonian structure, Darboux transformation and new soliton-like solutions of the Jaulent-Miodek (JM) hierarchy;(a) Based on the JM spectral problem, the integrable JM hierarchy is obtained, and two types of DTs of the hierarchy are constructed;(b) Via the symplectic-cosymplectic factorization, the Hamiltonian structure of JM hierarchy is obtained, and the hierarchy is proved to be completely integrable in the Liouville sense;(c) Through the two type of DTs, new soliton-like solutions are derived, which are all composed of the shock wave and bell-shaped soliton;(d) Propagation characteristics and interactions of the obtained solitons are graphically discussed.(5) Gauge transformation between the isospectral and first-order nonisospectral Kaup-Newell (KN) hierarchies.(a) The basic concept of gauge transformation is briefly intro-duced. The gauge transformation between the isospectral and first-order nonisospectral KN spectral problems is derived by introducing the isospectral and first-order non-isospectral AKNS spectral problems;(b) The equivalence between the isospectral and first-order nonisospectral KN hierarchies, as well as the first three representative equa-tions, is given;(c) By taking the isospectral and first-order nonisospectral KN systems for example, the direct gauge transformation between the corresponding isospectral and nonisospectral problems of the same system is investigated.(6) The applications of the bilinear method and multi-linear variable separation ap-proach in (2+1) dimensional dispersive long wave (DLW) equation.(a) Three types of the dependent variable transformations used in the bilinearization, as well as other de-pendent variable transformations, are introduced, and the corresponding types of NLEEs are also presented;(b) The multi-linear variable separation approach is used to investi-gate nonlinear localized excitations, and several common functions for localized solutions are briefly introduced;(c) The Painleve analysis is performed on the (2+1) dimensional DLW equation, and two types of expansion, i.e., one and two singular manifolds, are found. Correspondingly, two different forms of the dependent variable transformation are derived;(d) Via the two types of the dependent variable transformations, the DLW equation is respectively bilinearized and linearized, and further the analytic soliton solu-tions and nonlinear localized excitations are obtained correspondingly. Besides, figures are made to probe the soliton propagation characteristics and interactions.