Soliton Solutions And Their Dynamical Properties Of Several Nonlocal Coupled Nonlinear Schr(?)dinger-type Systems

In this dissertation,we mainly study several nonlocal and classical nonlinear Schr(?)dinger(NLS)-type integrable systems.We obtain different types of solutions of these systems,including soliton solutions,breather solutions,rogue wave solutions and period solutions,and investigate the interactions between different solutions and the evolutionary properties of some solutions.The first two parts of this dissertation study two nonlocal NLS-type systems.Ablowitz and Musslimani Phys.Rev.Lett.110(2013),064105 proposed a new nonlinear integrable equation iqt(x,t)+qxx(x,t)±2q2(x,t)q*(-x,t)=0,(2)where*presents the complex conjugate.They named it as nonlocal NLS equation and solved its Cauchy problem by the inverse scattering transform method.The usual nonlocal NLS equation is iqt(x,t)+qxx(X,t)±2q(x,t)??+?R(y-x)|q(y,t)|2dy=0,where R(x)is a real,localized and symmetric function.In nonlinear fiber model,R(x)is the response function of the nonlocal nonlinear Kerr-type media.But the nonlocal meaning of Eq.(2)is that the information of q(x,t)is not only related to the point(x,t)but also to the information at the space reverse point(-x,t).Because the nonlocal NLS equation(2)is PT-symmetric,i.e.after the parity transformation P(q)(x,t)=q(-x,t)and time reversal transformation T(q)(x,t)=q*(x,-t),the structure of Eq.(2)remains unchanged.The PT-symmetric systems are important contents in quantum mechanics and optics.So,Eq.(2)has attracted people's interests since it is proposed.Ma and Zhu J.Math.Phys.57(2016),083507 studied the gauge equivalence of the nonlocal NLS equation.Furthermore,Gadzhimuradov and Agalarov Phys.Rev.A 93(2016),062124 pointed out that the nonlocal NLS equation is gauge equivalent to a coupled Landau-Lifshitz equation.These results show that there are much importance in studying the nonlocal NLS equation.In chapter 2,we investigate an integrable nonlocal coupled NLS system.Khara and Saxena[J.Math.Phys.56(2015),032104]studied a nonlocal coupled NLS sys-tem,which can degenerate to the nonlocal Manakov system and nonlocal Mikhailov-Zakharov-Schulman(MZS)system with special conditions.Because Manakov system and MZS system are both famous coupled NLS systems,they asked a question“Whether the nonlocal Manakov system or nonlocal MZS system is integrable?”The nonlocal coupled NLS system that we study includes the nonlocal Manakov system and the nonlocal MZS system.We demonstrate the integrability of this nonlocal coupled NL-S system by proposing its Lax pair and infinitely many conservation laws.Thus we answer the question of Khara and Saxena.We construct the Darboux transformation for the integrable nonlocal coupled NLS system.By using Darboux transformation,we obtain several types of solutions of the nonlocal coupled NLS equations,and study the interaction between different solutions and the evolutionary properties of these so-lutions.Compared with the solutions of the nonlocal NLS equation,the solutions of the nonlocal coupled NLS are much richer,and the evolutionary properties of solutions are different.In chapter 3,we explore the reverse space-time nonlocal Sasa-Satsuma equation.Sasa-Satsuma equation can describe the propagation of femtosecond pulses in optical fibers and it is a high-order NLS equation.So the reverse space-time nonlocal Sasa-Satsuma equation can be viewed as a reverse space-time nonlocal high-order NLS equation.The reverse space-time nonlocal Sasa-Satsuma equation relates to a 3 x 3 spectral problem.Thought the integrable nonlocal equations relating to high-order spectral problems are proposed in other articles,but the solutions of these equations have not yet reported.We derive different types of solutions of the reverse space-time nonlocal Sasa-Satsuma equation with the Darboux transformation method,including periodic solutions,dark soliton solutions,W-shape soliton solutions,M-shape soliton solutions and breather solutions.We find these solutions are much richer than the solutions of the nonlocal NLS equation.In the last two parts of this dissertation,two integrable multi-component NLS-type systems are investigated.Because many physical systems have more than one components,such as multi-mode fiber,the Bose-Einstein condensate,etc.So the multi-component nonlinear integrable system is also one of the research hot spots in the theory of integrable systems.The solutions' structures for multi-component integrable system are much richer than those for the single component integrable system,and there are some differences in the interactions between different solutions.For example,there are some new types of soliton solutions for the two component NLS equations,including the breather solutions that are composed of bright and dark solitary waves,the fission breather solutions,and the boomeron-type solutions.There are inelastic collisions between bright solitary waves.In chapter 4,we study the(N+1)-component long-wave-short-wave resonance in-teraction(LSRI)equations.The LSRI equation can describe the resonance phenomenon between the capillary wave and the gravity wave in shallow water[J.Fluid Mech.79(1977),703].In addition,the LSRI equation has application in plasma physics[Pro-gr.Theor.Phys.56(1976),1719],nonlinear optics[Phys.Rev.Lett.100(2008),153905].Although there are a lot of works about solving the soliton solutions of the(N+1)-component LSRI equation,we still concern about whether there are other types of soliton solutions.We give the binary Darboux transformation of the(N+1)-component LSRI equations.Through the binary Darboux transformation,we obtain some exact solutions for the(2+1)-component LSRI equations.We not only get some new soliton solutions,including the new type breather solutions which are a combina-tion of bright and dark solitary wave,the fission breather solutions,the boomeron-type soliton solutions,but also acquire and do a classification analysis on the breather solu-tion and rogue wave solution.Our work further improves the study of multi-component LSRI equations.In chapter 5,we investigate a system obtained by coupling two-component NLS equations with the Boussinesq equation(the two-component NLS-Boussinesq coupled equations).The NLS-Boussinesq coupled equations can describe the bidirectional prop-agation low-frequency wave response and the high-frequency wave response of plasma when the propagation speed of the low-frequency wave is near the magnetosonic speed in the magnetic plasma field[J.Plasma Phys.39(1988),385].There are many s-tudies about the NLS-Boussinesq coupled equations,but hardly work with respect to the multi-component NLS-Boussinesq coupled equations.We obtain the bright-bright solitons,bright-dark solitons,dark-dark solitons,breathers and rogue wave solutions of the two-component NLS-Boussinesq coupled equations by using the Hirota bilinear method.We discuss the interaction between these solitary waves of the integrable case.We observe the inelastic collisions between bright solitary waves and the period-ical phenomenon in the two parallel-traveling bright-dark soliton solutions.These do not appear in the NLS-Boussinesq coupled equations.The breather solutions and rogue wave solutions of the integrable case are also given.The analysis of the modulation instability is discussed at last,which indicates that the appearances of the breathers and rogue waves have inseparable relationships with the plane wave instability.