Analytic Investigation On The Nonlinear Schrodinger-Type Equations In Some Fields Such As The Optical Fiber Communications

In recent decades all kinds of nonlinear phenomena have been one of research hot areas in physics. In theory, the essential features of dif-ferent kinds of nonlinear phenomena such as solitons and rogue waves are usually explored by analytic solutions of the corresponding nonlinear partial differential equations. As an class of nonlinear partial differen-tial equations, the nonlinear Schrodinger (NLS)-type equations appears in the studies of solitons and rogue waves of such fields as the optical fiber communications, maganetic materials, proteins, plasmas and fluids. Nowadays, the soliton and rogue wave’s notion has extended from physics and mathematics to engineering problems and natural phenomena. From the analytical point of view, several important NLS-type equations are investigated in this dissertation. For those equations, some soliton and rogue wave solutions are obtained which have certain theoretical value and can guide engineering practices to some extent. The main contents are as follows:Chapter1as the introduction starts with the connotation and sig-nificance of soliton, rogue wave and NLS-type equations, then introduces integrability of nonlinear partial differential equations in two senses and several mathematical methods involved. The main work and the organi-zation of this dissertation are also reported. In chapter2, under investigation is a NLS equation in the presence of the chirp and loss terms, which describes the averaged dispersion-managed (DM) fiber system with fiber losses and without the amplification effect at the end of one dispersion map. On the one hand, via the double Wron-skian technique N-soliton solutions are constructed from the correspond-ing Lax pair and verified to satisfy the corresponding bilinear equations. On the basis of the N-soliton solutions, it is analyzed that how the aver-aged DM soliton’s physical quantities such as the chirp, amplitude, veloc-ity, width and energy change. The effects of the chirp-loss coefficient on those physical quantities are given. The interacations between/among the averaged DM solitons are also discussed. Those results provide theoretical basis for realization of the optical information transmission with ultrahigh speed and high capacity. On the other hand, a hierarchy of rogue wave solutions on the background of continuous wave with the chirp and loss are obtained for this equation via the generalized Darboux transforma-tion. The effects of the chirp-loss coefficient on the optical rogue wave are discussed based on the first-order rogue wave solution. The first-, second-and third-order rogue wave solutions are illustrated. Those results could not only give theoretical basis for supercontinuum generation in a fiber and pulsed operation of passively mode-locked lasers, but also serve as references for the study of rogue waves in the ocean.In chapter3, a fourth-order dispersive NLS equation, which works as a model corresponding to a one-dimensional anisotropic Heisenberg fer-romagnetic spin chain with the octuple-dipole interaction, is investigated. Beyond the existing constraint, upon the introduction of an auxiliary function, bilinear forms and N-solition solutions are constructed with the Hirota method. With the help of the Wronskian representation of the aux-iliary function, N-soliton solutions in the form of double Wronskian are constructed and verified. Two Wronskian identities utilized in the process of verification are presented. From the N-soliton solutions, the expres-sions for the amplitude, width and velocity of the spin-wave-envelope soli-ton are given. The interactions between/among the spin-wave-envelope solitons are discussed with asymptotic analysis and graphics. Besides, the influence of the coefficient of higher-order magnetic interactions on the spin-wave-envelope soliton’s velocity and the bound-state period is analyzed. Those results may be significant to the microwave communica-tion systems and nonlinear signal processing devices.In chapter4, the dark soliton to explain the energy transfer in the proteins is studied motivated by the dark soliton describing the folded protein. The corresponding equation is a generalized fourth-order disper-sive NLS equation, which governs the nonlinear collective excitation in the alpha helical protein with higher-order amide-I vibrations and inter-actions. For the particular choice of parameters, a special case of that generalized NLS equation is presented. Painleve analysis is performed to prove that the special case is integrable. Through the introduction of an auxiliary function, bilinear forms and dark N-soliton solutions of the integrable case are constructed with the Hirota method. Based on the dark N-soliton solutions, the expressions for amplitude, width and veloc-ity of the dark-soliton are obtained. The head-on interaction between the two solitons, overtaking interaction between the two solitons, interaction between a moving soliton and a stationary one, interactions among three solitons are discussed with asymptotic analysis and graphics. It is also discussed how the coefficient of higher-order amide-I vibrations and in-teractions affects the dark-soliton velocity. The dark soliton in the alpha helical protein may be helpful to understand the mechanism of muscle contraction.Chapter5as the conclusion summarizes the contents and innovations of this dissertation and put forwards some further research work.