Investigation On Solitons And Rogue Waves For Certain Nonlinear Evolution Equations

In the reality world,the development of things is often affected by multi-ple factors rather than determined by the linear relationship formed by a single element.In such disordered,irregular and non-equilibrium systems,multiple variables interact with each other,resulting in the nonlinear phenomena.From the mathematical point of view,nonlinear phenomena can be described by the nonlinear evolution equations.With the aid of the mathematical research meth-ods,physical evolution process of the nonlinear models can be demonstrated more clearly,which is helpful for people to understand the development law and essential characteristics of many natural phenomena.We investigate certain nonlinear evolution euqations in such physical field as the nonlinear fiber optics,biology and marine dynamics analytically.Via the Hirota method and Darboux transformation,we derive the soliton solutions,rogue wave solutions and breather solutions.Base on such solutions,we study the propagation of the solitons,rogue waves and breathers in different nonlinear systems,as well as their interactions.The main contents in this paper are as follows:In Chapter 2,we investigate the coupled nonlinear Schrodinger equation-s with the third-order dispersion,which is also refered to as the coupled Hirota equations.The coupled Hirota equations have been used in describing the simul-taneous propagation of two ultrashort optical fields in the optical fiber,as well as the wave propagation in the marine dynamics.We consider the coupled Hirota e-quations in the case of the mixed regime and the defocusing-defocusing regime.With the help of the Darboux transformation,two types of the first-order and second-order localized wave solutions for the coupled Hirota equations are de-rived.Based on such solutions,moving breathers,Akhmediev breathers,multi-peak solitons and anti-dark solitons are obtained.Moreover,the breather-to-soliton conversion is studied.Elastic interactions between the breathers and dark solitons are studied.Additionally,inelastic interaction between the anti-dark soliton and dark soliton is shown.In Chapter 3,we investigate the three coupled Hirota system,which is ap-plied to model the long distance communication and ultrafast signal routing sys-tems governing the propagation of light pulses.With the aid of the Darboux dressing transformation,composite rogue wave solutions are derived.Nov-el spatial-temporal structures including the four-petaled structure for the three coupled Hirota system are exhibited.The composite rogue waves can split up into two or three single rogue waves.The corresponding conditions for the oc-currence of such phenomena are discussed and presented.Besides,the linear instability analysis is performed.In Chapter 4,we investigate the three coupled variable-coefficient nonlin-ear Schrodinger equations,which describe the amplification or attenuation of the picosecond pulse propagation in the inhomogeneous multi-component opti-cal fiber with the different frequencies or polarizations.Based on the Darboux dressing transformation,semirational rogue-wave solutions are derived.Semi-rational rogue waves,Peregrine combs and Peregrine walls are obtained and demonstrated.Additionally,the splitting behaviour of the semirational Pere-grine combs and attenuating phenomenon of the semirational Peregrine wall are exhibited.Effects of the group velocity dispersion,nonlinearity and fiber gain/loss are discussed according to the different fiber systems.In Chapter 5,we investigate the(3+1)-dimensional nonlinear Schrodinger equation,which describes the evolution of a slowly-varying wave packet enve-lope in the inhomogeneous optical fiber.With the Hirota method and symbolic computation,the bilinear form and dark multi-soliton solutions under certain variable-coefficient constraint are derived.Interactions between the different-type dark two solitons have been asymptotically analyzed and presented.Inter-action between the two periodic-type dark solitons is also presented.In Chapter 6,we investigate the generalized(2+1)-dimensional variable-coefficient Nizhnik-Novikov-Veselov equations in an inhomogenous medium,which are seen to describe the shallow water waves,lattice dynamics,ion-acoustic waves and plasma physics.Via the Hirota bilinear method and symbol-ic computation,we derive the bilinear forms,N-soliton solutions and Backlund transformations.Soliton evolution and interactions are graphically presented and analyzed.