| With the rapid development of science and technology,nonlinear integrable systems are playing an important role in more and more fields.Integrable systems and their nonlinear local wave solutions can be used to describe nonlinear phenomena in many fields such as oceanography,physics,biology,nonlinear optics,etc.In this paper,the generalized(m,N-m)-fold Darboux transformation,modulation instability,related hierarchy,Hamiltonian structure,infinitely many conservation laws,novel localized wave structures and their corresponding dynamical behaviors of several types of nonlinear integrable systems are studied on the basis of Lax integrability in the sense of 2×2 matrix linear spectrum problem.The main contents include the following three aspects:(1)The relations between modulational instability and localized wave excitations are studied for three continuous integrable systems including two(1+1)-dimensional and one(2+1)-dimensional systems.Based on the known Lax pair,the generalized(m,N-m)-fold Darboux transformations are constructed.Novel localized wave interaction structures among rogue wave,breather and semi-rational soliton of two(1+1)-dimensional systems are obtained,and novel localized wave interaction structures between Lump solution and breather of(2+1)-dimensional system are also given.The dynamic behavior of localized wavea is discussed through graphical analysis and numerical simulation with the help of Maple and Matlab.(2)The relations between modulational instability and localized wave excitations for two single component discrete integrable systems are studied,and the related hierarchy,Hamiltonian structure and Liouville integrability are given and established.Based on their known Lax pair,the discrete generalized(m,N-m)-fold Darboux transformations are constructed.Novel localized wave interaction structures among soliton,rogue wave,breather and semi-rational soliton are obtained.With the help of computer software Maple and Matlab,the evolution and propagation of localized waves are studied,and their dynamic behaviors are discussed by numerical simulation.(3)Soliton elastic interactions,modulation instability and novel localized wave structures for four kinds of coupled discrete integrable systems are investigated,and the related hierarchy and infinitely many conservation laws are discussed.The discrete generalized(m,N-m)-fold Darboux transformations are constructed on basis of their known Lax pair,different types of localized wave interaction structures are given.By asymptotic analysis and graphical analysis,the state expressions of the solitons are given before and after the collision,and their elastic interaction phenomena are discussed.With the aid of Matlab,the evolution and propagation stability of localized waves are studied via numerical simulation.This paper is divided into five chapters.The first chapter introduces the concept and development history of integrable system and localized waves,the main analytical methods of integrable system and the research background and content arrangement of this paper.Chapter 2 studies the modulation instability,localized waves and their dynamic properties of three continuous integrable systems,which mainly includes:(1)The related hierarchy of the modified self-steepening nonlinear Schr?dinger equation is constructed,the novel localized wave structures and their dynamic behavior are studied by the generalized(2,N-2)-fold Darboux transformation;(2)The novel localized wave structures and their dynamic behavior of the(1+1)-dimensional classical nonlinear Schr?dinger equation are studied by the generalized(2,N-2)-fold and(3,N-3)-fold Darboux transformations;(3)The novel localized wave interaction structures between Lump solution and breather of the(2+1)-dimensional KMN equation are studied the generalized(2,N-2)-fold Darboux transformation.Chapter 3 mainly investigates the modulation instability,localized waves and their dynamic properties,which mainly includes:(1)The novel localized wave structures and their dynamic behavior of discrete Ablowitz Ladik equation are studied by the discrete generalized(2,N-2)-fold and(3,N-3)-fold Darboux transformations;(2)The hierarchy associated with discrete Hirota equation is constructed,and the related Hamiltonian structure and Liouville integrability and established by using the trace identity,and different types localized wave interaction structures and their and dynamic behavior of this equation were studied by the discrete generalized(2,N-2)-fold Darboux transformation.Chapter 4 studies the modulation instability,soliton elastic interactions,localized wave structures and their dynamic evolution of four kinds of coupled discrete integrable systems,which mainly includes:(1)The hierarchy associated with the discrete reduced triangular-lattice ribbon equation is constructed,and bright-bright vector soliton elastic interaction,novel vector localized wave soliton interaction structures and their dynamic behavior are studied by discrete generalized(m,N-m)-fold Darboux transformation;(2)The novel dark-bright,bright-dark,bright-bright vector localized wave structures and their dynamic behavior of the reduced zigzag-runged ladder lattice equation are investigated onthe background of three types of different seed solutions by discrete generalized(m,N-m)-fold Darboux transformation;(3)The bright-bright vector soliton elastic interaction,novel vector localized wave soliton interaction structuresand their dynamic behavior are discussed for the discrete higher-order coupled Ablowitz-Ladik equation by discrete generalized(m,N-m)-fold Darboux transformation;(4)The dark-dark vector multi-soliton solutions and rational solutions of three discrete nonlinear differential-difference equations are investigated via the generalized(m,N-m)-fold Darboux transformation,respectively,the elastic phenomena of such dark solitons are discussed,and the numerical stability of soliton solutions for three equations in the same hierarchy are compared.The fifth chapter is the main conclusions,shortcomings and prospects of this paper. |
PDF Full Text Request