Analytic Investigation On Certain Nonlinear Evolution Equations In Physics

The investigation of nonlinear phenomena is one of hot research areas in physical science in recent years.In the fields of optical fiber,hydrody-namics and Bose-Einstein condensation,many nonlinear phenomena can be analyzed by studying the corresponding nonlinear evolution equation-s,such as nonlinear Schrodinger(NLS)-type equations and shallow water wave equations.The nonlinear wave phenomena described by these e-quations are consistent with the observations in nature and experiments,such as solitons and rogue waves.In this dissertation,the nonlinear wave solutions of certain equations are studied theoretically through analyti-cal methods,which will be helpful to understand and predict nonlinear waves,and play a valuable role in the future possible observations and applications.The main research can be summarized as follows:(1)Mixed-type vector solitons of the coupled Hirota systems in opti-cal fibers are investigated.With the help of the auxiliary functions,Hirota direct method and symbolic computation,firstly,the one-and two-bright-dark soliton solutions for the two types of 2-coupled Hirota system are obtained,and the properties of bright-dark solitons are studied.Then,the study of the 2-coupled Hirota system is extended to the 3-coupled and even N-coupled Hirota systems to discuss the properties and interactions of the multi-component mixed-type solitons.(2)Solitons and semi-rational rogue waves of the Sasa-Satsuma-type equations in optical fibers are investigated.For the scalar Sasa-Satsuma equation,based on the known bilinear forms,via the modified expanded formulas and symbolic computation,the generalized bright two-soliton solutions are constructed;Through classifying the interactions under d-iff erent parameter conditions,six cases of interactions between the two solitons are revealed via an asymptotic analysis,and the fact that the s-calar Sasa-Satsuma equation admits both the shape-preserving and shape-changing interactions between the bright two solitons is found.For the coupled Sasa-Satsuma equations,the Darboux-dressing transformation and matrix analysis are used to construct dark-bright one-soliton and semi-rational rogue-wave solutions;some rich nonlinear wave phenome-na are found,such as dark-bright breather-like solitons and coexistence between the breather-like soliton and double-peak rogue wave,and the stabilities of these nonlinear waves are analyzed by numerical simulation.(3)Under investigation are the nonautonomous rogue waves and vec-tor dark solitons of the NLS-type equations with variable coefficients.The Kadomtsev-Petviashvili(KP)hierarchy reduction and Gramian are used to construct higher-order rogue wave-like solutions for a nonautonomous NLS equation with external potentials and vector N-dark soliton solu-tions for a coupled NLS system with variable coefficients.Through the analytic and graphic analysis,the fact that nonautonomous rogue waves in the higher-order cases have more complex and richer features is found.For vector dark solitons,the existence and nondegenerate constraints are studied,and the influence of one and two solitons under different group ve-locity dispersion and amplification/absorption coefficients are discussed.(4)Study on nonautonomous vector solitons in the Bose-Einstein condensation.Similarly,via the KP hierarchy reduction and Gramian,the N-dark-dark,N-bright-dark and N-bright-bright soliton solutions for coupled Gross-Pitaevskii equations with a time-dependent external har-monic potential are constructed.The existence constraints of each type of vector solitons are obtained,and the properties of growth,decay and periodic oscillation of these matter-wave solitons are analyzed.(5)Study on a variable-coefficient generalized dispersive water-wave system.With the help of the Bell polynomials,bilinear forms,Backlund transformations and Lax pair,one-and two-soliton solutions for the sys-tem are constructed.Based on the asymptotic and graphic analysis,the fission and fusion phenomena between two solitons are studied,and the fact that soliton interactions may be elastic or inelastic under the differ-ent variable coefficients is found.These results may help researchers to understand the nonlinear dispersive water-wave equations with variable coefficients.