Analytic Study On Several Nonlinear Models In Optical Fiber Communications,Fluids And Other Fields

With the development of the nonlinear science,solving the nonlinear evolution equations has become an important research topic.Particularly,nonlinear evolution equations admit several types of analytical solutions,such as soliton solutions,rogue-wave solutions,lump solutions and so on,and the corresponding nonlinear waves have been used to describe the nonlinear phenomena in optical fiber communications,fluids and other fields.In this dissertation,with the aid of analytic methods,we investi-gate several Schr?dinger equations in nonlinear optical fibers and several Kadomtsev-Petviashvili(KP)-type equations in fluids,and then we research the propagations of soliton,rogue wave and lump wave.The main contents of the dissertation can be sum-marised as:(1)In Chapter 1,we introduce several nonlinear waves represented by soliton,rogue wave and lump wave as well as their research progress.Analytical methods used in this dissertation to solve the nonlinear evolution equations are briefly displayed.(2)In Chapter 2,investigated is a variable-constant coupled nonlinear Schr?dinger system.By virtue of the Hirota method,we construct one-and two-soliton solution-s.Based on such solutions,we show the elastic and inelastic interactions between two solitons with the help of figures.Furthermore,we derive the bilinear B?cklund transfor-mation for the system.(3)In Chapter 3,under investigation is a fourth-order variable-coefficient nonlinear Schr?dinger equation.Firstly,we find the integral aspect of the equation,and construct the bilinear form for the equation via an auxiliary function.Then,we construct dark one-and two-soliton solutions for the equation via the Hirota method.We graphically study the solitons with the variable coefficients of the group-velocity dispersion,third-order dispersion and fourth-order dispersion,respectively;With the different choices of the variable coefficients,we obtain the parabolic,periodic and V-shaped dark solitons.We also perform the asymptotic analysis and prove that the interaction between two solitons is elastic.(4)In Chapter 4,investigated is the generalized nonlinear Schr?dinger equation,which describes the amplification or absorption of pulses propagating in a monomode optical fiber.By employing the KP hierarchy reduction,we obtain the rogue waves based on rogue-wave solutions in terms of the Gramian under integral constraint.The eye-shaped distribution density first-order rogue wave and the second-order rogue waves with the highest-peak amplitude and with the triple-peak structure are displayed.We also study the effects of group-velocity dispersion,nonlinearity and amplification/absorption coefficients on the rogue waves with the help of figures.(5)In Chapter 5,under investigation is a(3+1)-dimensional potential Yu-Toda-Sasa-Fukuyama equation.Based on the bilinear forms,we obtain the bilinear B?cklund transformation and the N-soliton solutions in terms of the Gramian.Elastic and inelastic interactions between two solitons are shown.Furthermore,based on the Pfaffianization,Wronski-and Gram-type Pfaffian solutions for a Pfaffianized coupled(3+1)-dimensional potential Yu-Toda-Sasa-Fukuyama system are derived.(6)In Chapter 6,a(3+1)-dimensional B-type KP equation and a(3+1)-dimensional generalized B-type KP equation are investigated.For the(3+1)-dimensional B-type KP equation,through the Hirota method and the extended homoclinic test technique,we obtain the breather type kink soliton solutions and breather rational soliton solutions.Rogue wave solutions are also derived.Furthermore,with the aid of the Hirota method and symbolic computation,lump solitons are also obtained;for the(3+1)-dimensional generalized B-type KP equation for water waves is investigated.Through the KP hier-archy reduction,rogue waves and lump solitons based on the solutions in terms of the Gramian.Interactions among several first-order rogue waves/lump solitons are present-ed.(7)In Chapter 7,a(2+1)-dimensional Davey-Stewartson system is studied,which describes the surface water wave packets of finite depth.Based on the KP hierarchy reduction,we derive two types of the solutions in terms of the Gramian,including the solutions containing dark solitons and rogue waves/lump solitons,and the solutions containing certain solitons and breathers.We find the interactions between the rogue waves/lump solitons and dark solitons are elastic.Besides,we observe the interactions between the dark solitons and three types of breathers.(8)In Chapter 8,under investigation is a(3+1)-dimensional modified Korteweg-de Vries-Zakharov-Kuznetsov equation.With the Hirota method,we derive the one-,two-and three-soliton solutions for such an equation.It is shown that the interactions between the solitons are elastic.We also graphically study the solitons related with the coefficient of the cubic nonlinearity.Such equation has been used to describe the ion-acoustic waves in a magnetized plasma.(9)Finally,in Chapter 9,the dissertation is summarized and the future research contents are proposed.