Analytic Investigation On The Nonlinear Models With The Symbolic Communication

Nonlinear waves such as solitons,breathers and rogue waves have been seen in such fields as the optical fiber,Bose-Einstein condensation,plasmas physics and fluids,and described by the nonlinear Schrodinger-type equations as a kind of nonlinear evolution equations.Solitons have been verified to be formed as a result of the balance between the dispersion and nonlinear effects.Breathers also form an important part of nonlinear wave systems,manifesting themselves as a localized spatial or temporal structure that exhibits oscillatory behaviour.Rogue waves,reported to be localized in the both space and time,have been found to appear from nowhere and disappear without any trace,here,from the analytical point of view and with the help of symbolic computation,the solitons,breathers and rogue waves in the optical fiber and other fields are investigated based on the nonlinear evolution equations.The main contents are as follows:(1)Via the Hirota method and symbolic computation,We report the existence of the lump wave,breather wave and rogue wave solutions for a(3+1)-dimensional generalized Kadomtsev-Petviashvili equation in fluid.Firstly,We derive the lump wave,breather wave and rogue wave solutions.Then,the properties of lump,breather and rogue waves are exhibited graphically.(2)Under investigation is a coupled variable-coefficient higher-order nonlinear Schrodinger system,which describes the simultaneous propa-gation of optical pulses in an inhomogeneous optical fiber.Based on the Lax pair and binary Darboux transformation,we present the nondegener-ate N-dark-dark soliton solutions.With the graphical simulation,soliton propagation and interaction are discussed with the group velocity disper-sion and fourth-order dispersion effects,which affect the velocity but have no effect on the amplitude(3)Under investigation in this paper are the coupled variable-coefficient fourth-order nonlinear Schrodinger equations,which describe the simul-taneous propagation of optical pulses in an inhomogeneous optical fiber.Based on the Lax pair,we construct the Darboux transformation,and obtain the the localized wave solutions including the breather solutions,rogue wave solutions and interactions of the localized waves.Influences of the group velocity dispersion and fourth-order dispersion on such so-lutions are analyzed:(i)There exist two types of breathers,in which the group velocity dispersion and fourth-order dispersion have no effect on the wave structure and trajectory of the one type,while have impact on those of the other,(ii)Four-petaled and dark-bright rogue waves are de-rived.(iii)Interactions between the breathers and dark-dark solitons axe demonstrated in two cases:In one case,interactions have no influence on the directions and phase shifts of the breathers,but affect the phase shifts and wave structures of the dark solitons.In the other case,breathers and solitons both produce certain phase shifts during the interaction,where the group velocity dispersion and fourth-order dispersion affect the wave structures of the breathers and solitons.