Nonliner Wave Solutions And Their Properties For Certain Equations In The Ablowitz-Kaup-Newell-Segur Hierarchy

Nonlinear evolution equations can describe the evolution of nonlinear phenomena in the fields of optical fiber optics,hydrodynamics and plasma physics.Nonlinear wave solutions of such equations are analytical and describe the nonlinear waves including solitons,breathers,rogue waves and the nonlinear superposition of several nonlinear waves.In this dissertation,based on four nonlinear evolution equations in the AblowitzKaup-Newell-Segur hierarchy,i.e.,a coupled nonlinear Schrodinger equation,a coupled Hirota equation,a coupled nonlinear Schrodinger equation with the negative coherent coupling and a three-component coupled nonlinear Schrodinger equation,we investigate certain nonlinear wave solutions and their properties,and obtain some results in the study of superregular breathers,degenerate nonlinear waves and Darboux transformation induced via a rank-two projection matrix.Main content of this dissertation is presented as follows:In chapter 1,we introduce the background knowledge of Ablowitz-Kaup-NewellSegur hierarchy,Darboux transformation,nonlinear wave and asymptotic analysis,describe the research progress of superregular breather,degenerate nonlinear wave and Darboux transformation,and point out the research content and structure of this dissertation.In chapter 2,we study the superregular solution for a coupled nonlinear Schrodinger equation,where the definition of superregular solution is generalized from the definition of superregular breather solution in the scalar field described via the nonlinear Schrodinger equation.On the nonzero-zero(or proportional nonzero-nonzero)background,regular solutions are the solutions which describe the regular nonlinear waves,and regular nonlinear waves are the waves located in a finite t domain and not perturbing the background with t being big enough;superregular solutions are a subset of regular solutions,and describe the nonlinear superposition of breathers and dark-bright(or breather-like)solitons developing from the dark-bright(or breather-like)solitons with the small perturbation at a certain z,where z and t denote the evolution dimension and temporal distribution dimension,respectively.On the nonzero-zero background,superregular solutions are constructed in three cases:trivial case,a pair of breathers case and single breather case,where the perturbation for the first case is on the nonzerozero background,but the perturbation for the latter two cases is on the dark-bright soliton.Then,other superregular solutions could be constructed according to the analyses for above three cases.We demonstrate that regular and superregular solutions on the nonzero-zero background could be transformed into the regular and superregular solutions on the proportional nonzero-nonzero background via the orthogonal transformation,respectively,and vice versa.In chapter 3,we investigate the superregular breather solution for a coupled nonlinear Schr(?)dinger equation with the negative coherent coupling.We construct the Darboux dressing transformation and the Nth-order breather solutions,where the positive integer N denotes the number of iterative times.We discuss the limits of the ratios between the Nth-order breather solutions and seed solutions.We introduce the condition to divide the first-order breathers into the trivial and nontrivial cases.For the latter case,besides deriving the single-hump and double-hump breathers,we analyse whether the breathers could be the kink-type breathers based on the limits derived above.Based on the second-order breathers,superregular breathers are derived,where each superregular breather is the nonlinear superposition of two(1)kink-type,(2)singlehump or(3)double-hump quasi-Akhmediev breathers.After the interaction,profiles of two quasi-Akhmediev breathers change for case(1)but keep unchange for case(2)or case(3).In chapter 4,we discuss the degenerate dark-bright soliton solution for a coupled nonlinear Schr(?)dinger equation.We construct the(N,m)-generalized Darboux transformation to derive the Nth-order solutions,where the positive integer m denotes the number of distinct spectral parameters.We search for such conditions to be satisfied that the Nth-order solutions describe the nonlinear superposition of degenerate dark-bright solitons and nondegenerate dark-bright solitons,and introduce a method to perform the asymptotic analyses on the degenerate dark-bright soliton solutions.We find that a degenerate dark-bright soliton is the nonlinear superposition of several dark-bright solitons possessing the same profile.Those dark-bright solitons are reflected during the interaction,and their velocities are z-dependent except that one of the velocities would become z-independent under the certain condition.It should be noticed that the distribution of those velocities is asymmetric under the certain condition.Interaction between a degenerate dark-bright soliton and a degenerate/nondegenerate dark-bright soliton is elastic,and the bound-state dark-bright soliton may be formed during the interaction,while the velocity of bound-state dark-bright solitons may be z-dependent or z-independent.In chapter 5,we investigate the degenerate fundamental nonlinear wave solution on the nonzero-zero background for a coupled Hirota equation.Iterating the existing Darboux transformation once,we derive the fundamental nonlinear wave solutions on the nonzero-zero background(except the semirational case),and find the condition to divide the fundamental nonlinear waves into three-branch and two-branch cases,where the three-branch fundamental nonlinear wave is the nonlinear superposition of a breather and two dark-bright solitons,and the two-branch fundamental nonlinear wave is the nonlinear superposition of a breather and a dark-bright soliton.The reason why the three-branch case is transformed into the two-branch case is not that the breather branch is transformed into the Kuznetsov-Ma breather branch,but that the velocities of three branch become equal.Iterating the existing Darboux transformation at the same spectral twice,we obtain the fundamental nonlinear wave solution in the degenerate case.We point out that the degenerate three-branch fundamental nonlinear wave is the nonlinear superposition of a pair of breathers and two pairs of dark-bright solitons,as well as each pair of nonlinear waves possess the same profile and the symmetry trajectories,while the degenerate two-branch fundamental nonlinear wave is the nonlinear superposition of a breather,a dark-bright soliton and a two-branch fundamental nonlinear wave,as well as the trajectories of such three nonlinear waves are asymmetry.In chapter 6,we investigate the Darboux transformation induced via a rank-two projection matrix for a three-component coupled nonlinear Schr(?)dinger equation.We construct a Darboux transformation induced via a rank-two projection matrix,and discuss the difference between it and the Darboux transformation induced via a rank-one projection matrix reported before,and also the difference between it and the generalized Darboux transformation based on the latter Darboux transformation.Based on the Darboux transformation induced via a rank-two projection matrix,we construct the(N,m)-generalized Darboux transformation and the Nth-order solution.We divide the fundamental nonlinear waves on the nonzero-zero-zero background into three cases,and discuss their properties through different asymptotic analyses.We further discuss such fundamental nonlinear waves in the degenerate case.On the one hand,nonlinear waves mentioned above are more useful to understand the wave evolution in the vector field described via the three-component coupled nonlinear Schr(?)dinger equation,because the solutions used to describe them could not be expressed via the solutions for nonlinear Schr(?)dinger equation or coupled nonlinear Schr(?)dinger equation.On the other hand,solutions used to describe above nonlinear waves can not be constructed via the Darboux transformation induced via a rank-one projection matrix.Thus,when three-component coupled equations are investigated,Darboux transformation induced via a rank-two projection matrix are more helpful than the ones induced via a rank-one projection matrix.In chapter 7,we summarize the study in this dissertation,and introduce the researching direction in the future.