| In the field of nonlinear science,solitons,breathers and rogue waves are three kinds of important nonlinear localized waves,which have a wide range of potential applications in physical fields such as nonlinear optics,water waves,plasma,BoseEinstein condensate and so on.Based on the known Lax pairs as the main line,some novel localized wave structures and their associated parameter controls for several classes of continuous and semi-discrete nonlinear integrable equations are investigated by using the generalized Darboux transformation.The main research covers the following five aspects:(1)Various soliton solutions and their dynamical behavior of the semi-discrete higher-order m Kd V equation are investigated.Based on the known 2 ?2 Lax pair,the discrete generalized(m,N-m)-fold Darboux transformation for this discrete system is constructed.Then,the various discrete soliton solutions such as the usual bell-shaped,anti-bell soliton-shaped,rational dark soliton and their mixed soliton solutions are derived.The limit states of diverse soliton solutions are analyzed via the asymptotic analysis technique.The elastic interaction phenomena and physical characteristics are discussed and illustrated graphically.Numer ical simulations are used to display the dynamical behaviors of some soliton solutions.(2)Two types of complex short pulse equations(continuous and semi-discrete cases)and a type of semi-discrete complex coupled dispersionless equation are investigated.The continuous limit of the semi-discrete complex short pulse equations is discussed,and some connections with the continuous complex short pulse equations are established.The relationship between the complex short pulse equation and the coupled dispersionless equation is also established through the Hodograph transformation.On the one hand,based on the transformed 2 ?2 Lax pairs,the continuous and the discrete generalized(m,N-m)-fold Darboux transformation are constructed for the continuous and discrete complex short pulse equations,respectively,which can generate three types of novel locationcontrollable localized wave solutions,including loop rogue waves,loop periodic waves and their loop mixed interaction structures;on the other hand,the modulation instability analysis of the semi-discrete complex coupled dispersionless equation is used to reveal the possible parameter regions for different local ized waves,and the discrete generalized(m,N-m)-fold Darboux transformation is constructed to give various novel localized wave structures including bright-dark rogue waves,doublepeaked double-valley rogue waves,periodic waves and mixed interaction structures.(3)The novel localized wave solutions of the two types of Wadati-KonnoIchikawa equations are investigated.Based on the known Hodograph transformation,we give an alternative two-component nonlinear system of the first type WadatiKonno-Ichikawa equation.The Hodograph transformations of the second type Wadati-Konno-Ichikawa equations are also investigated by means of its conservation laws,and a coupled nonlinear system of this equation in another coordinate system is given.Based on their known 2?2 order Lax pair,the generalized(m,N-m)-fold Darboux transformation for two types of transformed nonlinear systems are constructed,from which various novel position-controllable localized wave solutions are investigated including: the high-order rogue waves,periodic wave solutions and mixed interaction solutions with the smooth,singular and singular loop structures of the first type Wadati-Konno-Ichikawa equation are studied;based on the new Hodograph transformation,the soliton solutions,semirational solutions and rogue waves solutions for the second type Wadati-KonnoIchikawa equations are investigated,and the reasons for the emergence of the singular structures for the one-soliton and first-order rogue waves solutions are analyzed.(4)The matter rogue wave,periodic wave and their mixed interaction patterns in an F =1 spinor Bose-Einstein condensate for the three-component GrossPitaevskii equations are investigated.Firstly,the modulation instability analysis is used to disclose the possible parameter regions for the existence of different localized waves,then,the iterative generalized Darboux transformation based on known 4?4Lax pair is constructed,and various novel localized wave structures are given including the bright-dark-bright structure of rogue waves,periodic waves and their mixed interaction structures.Finally,the asymptotic states of higher-order rouge waves at parametric infinity are predicted by means of the large parameter asymptotic technique.(5)Novel localized wave solutions of the three-component synchronization Ablowitz-Ladik systems are investigated.Firstly,the modulation instability is used to reveal the possible parameter regions for the existence of different localized waves,then the generalized(m,N-m)-fold Darboux transformation is constructed based on its known 6?6 Lax pair,from which various position-controllable novel localized wave solutions are given including oscillatory rogue waves with multipeak and multi-valley,circular rogue wave s,periodic waves and their mixed interaction solutions.Finally,the numerical simulations are used to display the dynamical behaviors of some localized wave solutions.This paper is divided into seven chapters.Chapter 1 introduces the historical background,generation mechanism of localized waves,some methods for solving nonlinear integrable equations,the research background and structure arrangement of this paper.Chapter 2 investigates various soliton solutions,parameter control and their dynamical behavior of the semi-discrete high-order m Kd V equation via the generalized(m,N-m)-fold Darboux transformation.Chapter 3 investigates novel localized wave solutions and parameters control for three classes of nonlinear integrable equations via the generalized(m,N-m)-fold Darboux transformation including:(1)loop rogue waves,loop periodic waves and their mixed interaction solutions for the continuous complex short pulse equation;(2)discrete loop rogue waves,loop periodic waves and their mixed interaction solutions for the semidiscrete complex short pulse equation;(3)the modulation instability,bright-dark rogue waves,double-peaked double-valley,and their mixed interaction solutions for the semi-discrete complex coupled dispersionless equation.Chapter 4 investigates novel localized wave solutions and parameter control for two types of WadatiKonno-Ichikawa equations via the generalized(m,N-m)-fold Darboux transformation including:(1)high-order rogue wave,periodic wave and their mixed interaction solutions with smooth,singular and singular loop structures for the first type Wadati-Konno-Ichikawa equation;(2)the Hodograph transformation,conservation laws,soliton,semi-rational solutions and rogue wave solutions with smooth,singular and singular loop structures for the second type Wadati-KonnoIchikawa equation.Chapter 5 investigates the novel position controllable localized wave patterns and parameter control of the three-component Gross-Pitaevskii equation,the double-peaked rogue wave patterns and mixed interaction patterns are obtained by using the generalized iterative Darboux transformation,and the largeparameter asymptotic analysis of the higher-order rogue waves is performed.Chapter 6 mainly studies the novel localized waves and position control of the threecomponent Ablowitz-Ladik synchronization systems by constructing the generalized(m,N-m)-fold Darboux transformation,and the oscillatory rogue waves,circular rogue waves,periodic waves and their mixed interaction solutions are obtained.Chapter 7 presents the conclusions,shortcomings and prospects of this paper. |
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