Investigations On Several Nonlinear Models In Optical Fibers And Other Fields

The nature of the world is nonlinear.The diversity of natural and social phenomena determines the diversity of nonlinear phenomena.Nonlinear phenomena such as solitons,breathers,rogue waves and lump waves in optical fiber communication and other physical fields can be described via nonlinear evolution equations.Based on the nonlinear evolution equations,several kinds of nonlinear models in optical fiber communication and other physical fields are analyzed and discussed via Darboux transformation,generalized Darboux transformation and Hirota method.The propagation characteristics of solitons,breathers,rogue waves and lump waves in these nonlinear models are studied,and the interactions between them are further analyzed.The structure and main arrangement of this paper are shown as follows:In chapter 1,we introduce the backgrounds of symbolic computation and nonlinear science.Nonlinear wave is one of the main research contents in nonlinear science.We briefly introduce the developments of nonlinear waves via taking solitons,rogue waves,breathers and lump waves as examples,and present the methods used to study nonlinear waves in this paper,including Darboux transformation,generalized Darboux transformation and Hirota method.Finally,the main work and structures of this paper are given.In chapter 2,we discuss a coupled variable-coefficient cubic-quintic nonlinear Schr?dinger system,which can be used to describe the effects of quintic nonlinearity for the ultrashort optical pulse propagation.By virtue of the Darboux transformation and generalized Darboux transformation methods,we construct the second-and third-order vector breather solutions for a coupled variable-coefficient cubic-quintic nonlinear Schrodinger system.We take a gauge transformation to convert a Lax pair into an Ablowitz-KaupNewell-Segur system.In addition,we construct the Nth-order Darboux transformation,where N is a positive integer.Applying Taylor expansion,we give the Nth-order generalized Darboux transformation.Based on the Nth-order generalized Darboux transformation method,we obtain the second-and third-order vector breather solutions for the new system.According to the gauge transformation,we derive the second-and third-order vector breather solutions for the original system.In chapter 3,a three-coupled variable-coefficient nonlinear Schr?dinger system,which models the attenuation or amplification of the picosecond pulses in an inhomogeneous multicomponent optical fiber with different polarisations or frequencies,is researched.We give a new Lax pair,which is different from those in the existing literatures.Based on our Lax pair,we construct the Nth-order generalized Darboux transformation,and then obtain the first-and second-order vector breathers solutions.In addition,we show the propagation of the first-order vector breathers with wave-shaped,and the second-order vector breathers with S-typed.Besides,the first-and second-order vector breathers in an inhomogeneous multicomponent optical fiber with different polarisations or frequencies are shown when we take the nonlinear and group velocity dispersion coefficients as trigonometric functions.In chapter 4,we focus on a coupled fourth-order nonlinear Schr?dinger system,which describes the ultrashort optical pluses in a birefringent optical fiber.Using the existing Lax pair,we derive the two-and three-soliton solutions for the system according to the generalized Darboux transformation method.We graphically display(1)the elastic interactions between/among the two/three solitons,where amplitudes of the solitons remain unchanged;(2)the inelastic interactions between/among the two/three solitons,where amplitudes of the solitons change;(3)the bound states among the three solitons;(4)the higher-order linear and nonlinear effects on the polarization components of the electric field.In chapter 5,we investigate a three coupled variable-coefficient nonlinear Schr?dinger system,which describes the amplification or attenuation of the picosecond pulses in an inhomogeneous multicomponent optical fiber with different frequencies or polarizations.Based on the existing Lax pair,we construct the generalized Darboux transformation and give the second-order semirational rogue-wave solutions,which represent the slowly varying envelopes of optical modes,under the constraints of fiber gain/loss,nonlinearity and group velocity dispersion coefficients.We display the influences of nonlinearity and group velocity dispersion coefficients.Finally,we analyze the modulation instability of the system via the linear stability.In chapter 6,we study a matrix Lakshmanan-Porsezian-Daniel equation in a Heisenberg ferromagnetic spin chain.Based on the Darboux transformation method and the generalized Darboux transformation method,we construct the Nth-order Darboux transformation and the Nth-order generalized Darboux transformation of the equation.Go a step further,we give the first-and second-order vector breather solutions.We show the propagation of the first-order vector breathers including two peaks breathers,bright circular breathers and dark circular breathers.Besides,we display the propagation of the second-order vector breathers including two crossed linear breathers,a linear breather intersects with an S-typed breather,and a linear breather overlaps with a circular breather.Finally,the influences of the higher-order linear and nonlinear effects coefficients on the propagation of first-and second-order vector breathers are also discussed.In chapter 7,we discuss a(3+1)-dimensional generalized Kadomtsev-Petviashvili equation in fluid mechanics.We derive the mixed lump-stripe waves solutions by virtue of the symbolic computation.We observe the fission and fusion phenomena between the lump and one-stripe wave through the mixed-stripe wave solutions.We graphically present the interactions between a bright lump and two bright solitons,and a dark lump and two dark solitons via the mixed lump-stripe solutions.Besides,we analyze the influences of the coefficients on the mixed lump-stripe waves.In chapter 8,we summarize the work of this dissertation,and point out the shortcomings.In addition,we give the future research directions.