Rogue wave solutions for the coupled cubic–quintic nonlinear Schrödinger equations in nonlinear optics

A generalized Darboux transformation for the coupled cubic–quintic nonlinear Schrödinger equation is constructed by the Darboux matrix method. As applications, the Nth-order rogue wave solutions of the coupled cubic–quintic nonlinear Schrödinger equation have been obtained. In particular, the dynamics of the general first- and second-order rogue waves are discussed and illustrated through some figures.     

On Nth-order rogue wave solution to nonlinear coupled dispersionless evolution equations

The generalized Darboux transformation is applied to obtain Nth-order rogue wave solution of nonlinear coupled dispersionless evolution equations (NLCDEE). In particular, the interesting structures for these solutions from first to third order rogue wave are shown.     

Rogue wave solutions for a generalized nonautonomous nonlinear equation in a nonlinear inhomogeneous fiber

In this paper, we investigate a generalized nonautonomous nonlinear equation which describes the ultrashort optical pulse propagating in a nonlinear inhomogeneous fiber. By virtue of the generalized Darboux transformation, the first- and second-order rogue-wave solutions for the generalized nonautonomous nonlinear equation are obtained, under some variable–coefficient constraints. Properties of the first- and second-order rogue waves are graphically presented and analyzed: When the coefficients are all chosen as the constants, we can observe the some functions, the shapes of wave crests and troughs for the first- and second-order rogue waves change. Oscillating behaviors of the first- and second-order rogue waves are observed when the coefficients are the trigonometric functions.     

New Rogue Wave Solutions of (1+2)-Dimensional Non-Isospectral KP-II Equation

The generalized binary Darboux transformation for the (1+2)-dimensional non-isospectral KP-II equation is presented. Moreover, as a direct application, the new rogue wave solutions for the (1+2)-dimensional non-isospectral KP-II equation are constructed by the generalized binary Darboux transformation.     

Rogue Wave Solutions for Nonlinear Schrödinger Equation with Variable Coefficients in Nonlinear Optical Systems

In this paper, the generalized Darboux transformation is constructed to variable coefficient nonlinear Schrödinger (NLS) equation. The N-th order rogue wave solution of this variable coefficient NLS equation is obtained by determinant expression form. In particular, we present rogue waves from first to third-order through some figures and analyze their dynamics.     

生长及耗散波色-爱因斯坦凝聚中的怪波

利用达布变换法(Darboux transformation),解析的研究了生长及耗散波色-爱因斯坦凝聚(BEC)中的怪波.通过降维和无量纲化,将描述BEC的Gross-Pitaevskii (GP)方程转化成一维无量纲非线性薛定谔方程.利用达布变换,得到了一维非线性薛定谔方程的怪波解析解.根据解析结果,数值模拟了生长及耗散BEC中怪波的性质.结果表明,BEC中出现了一种典型的双洞怪波,并且BEC生长会延缓怪波的消失,而BEC的耗散会加速怪波的消失.     

Solitary waves and rogue waves in a plasma with nonthermal electrons featuring Tsallis distribution

In this Letter, we discuss the electron acoustic (EA) waves in plasmas, which consist of nonthermal hot electrons featuring the Tsallis distribution, and obtain the corresponding governing equation, that is, a nonlinear Schrödinger (NLS) equation. By means of Modulation Instability (MI) analysis of the EA waves, it is found that both electron acoustic solitary wave and rogue wave can exist in such plasmas. Basing on the Darboux transformation method, we derive the analytical expressions of nonlinear solutions of NLS equations, such as single/double solitary wave solutions and single/double rogue wave solutions. The existential regions and amplitude of solitary wave solutions and the rogue wave solutions are influenced by the nonextensive parameter q and nonthermal parameter α. Moreover, the interaction of solitary wave and how to postpone the excitation of rogue wave are also studied.     

Solutions of Two Kinds of Non-Isospectral Generalized Nonlinear Schrodinger Equation Related to Bose-Einstein Condensates

Two non-isospectral generalized nonlinear Schrodinger （ONLS） equations, which are two important models of nonlinear excitations of matter waves in Bose-Einstein condensates, are studied. Two novel transformations are constructed such that these two GNLS equations are transformed to the well-known nonlinear Schr6dinger （NLS） equation, which is an isospectral equation. Therefore, once one solution of the NLS equation is provided, we can immediately obtain one solution for two ONLS equations by these transformations. Thus it is unnecessary to solve these two non-isospectral GNLS equations directly. Soliton solutions and periodic solutions are obtained for them by two transformations from the corresponding solutions of the NLS equation, which are generated by Darboux transformation.     

Rogue Waves of the Higher-Order Dispersive Nonlinear Schrödinger Equation

In this paper,the rogue waves of the higher-order dispersive nonlinear Schrödinger (HDNLS) equation are investigated,which describes the propagation of ultrashort optical pulse in optical fibers.The rogue wave solutions of HDNLS equation are constructed by using the modified Darboux transformation method.The explicit first and second-order rogue wave solutions are presented under the plane wave seeding solution background.The nonlinear dynamics and properties of rogue waves are discussed by analyzing the obtained rational solutions.The influence of little perturbation ε on the rogue waves is discussed with the help of graphical simulation.     

Localized waves in three-component coupled nonlinear Schrdinger equation

We study the generalized Darboux transformation to the three-component coupled nonlinear Schr ¨odinger equation.First-and second-order localized waves are obtained by this technique.In first-order localized wave,we get the interactional solutions between first-order rogue wave and one-dark,one-bright soliton respectively.Meanwhile,the interactional solutions between one-breather and first-order rogue wave are also given.In second-order localized wave,one-dark-one-bright soliton together with second-order rogue wave is presented in the first component,and two-bright soliton together with second-order rogue wave are gained respectively in the other two components.Besides,we observe second-order rogue wave together with one-breather in three components.Moreover,by increasing the absolute values of two free parameters,the nonlinear waves merge with each other distinctly.These results further reveal the interesting dynamic structures of localized waves in the three-component coupled system.     

Characteristics of Rogue Waves on a Soliton Background in the General Coupled Nonlinear Schrödinger Equation

Under investigation in this work is the general coupled nonlinear Schrödinger (gCNLS) equation, which can be used to describe a wide variety of physical processes. By using Darboux transformation, the new higher-order rogue wave solutions of the equation are well constructed. These solutions exhibit rogue waves on a multi-soliton background. Moreover, the dynamics of these solutions is graphically discussed. Our results would be of much importance in enriching and predicting rogue wave phenomena arising in nonlinear and complex systems.     

Higher-Order Rogue Wave Pairs in the Coupled Cubic-Quintic Nonlinear Schrödinger Equations

We study some novel patterns of rogue wave in the coupled cubic-quintic nonlinear Schrödinger equations. Utilizing the generalized Darboux transformation, the higher-order rogue wave pairs of the coupled system are generated. Especially, the first-and second-order rogue wave pairs are discussed in detail. It demonstrates that two classical fundamental rogue waves can be emerged from the first-order case and four or six classical fundamental rogue waves from the second-order case. In the second-order rogue wave solution, the distribution structures can be in triangle, quadrilateral and ring shapes by fixing appropriate values of the free parameters. In contrast to single-component systems, there are always more abundant rogue wave structures in multi-component ones. It is shown that the two higher-order nonlinear coefficients ρ₁ and ρ₂ make some skews of the rogue waves.     

玻色-爱因斯坦凝聚体中的怪波操控

利用Darboux变换法,解析地研究了玻色-爱因斯坦凝聚体(BEC)中的怪波.结果表明:当谱参数等于非线性系数时,BEC中形成一种新型的单洞怪波;而当谱参数小于非线性系数时,BEC中出现双洞怪波.进一步地,怪波的出现位置可通过调节周期性势阱的驱动频率和强度来控制.此外,随着原子间相互作用的减小,怪波的最高幅度也随之降低.相关结果可为预防怪波的危害提供帮助.     

玻色-爱因斯坦凝聚体中的怪波操控

利用Darboux变换法, 解析地研究了玻色-爱因斯坦凝聚体(BEC)中的怪波. 结果表明: 当谱参数等于非线性系数时, BEC中形成一种新型的单洞怪波; 而当谱参数小于非线性系数时, BEC中出现双洞怪波. 进一步地, 怪波的出现位置可通过调节周期性势阱的驱动频率和强度来控制. 此外, 随着原子间相互作用的减小, 怪波的最高幅度也随之降低. 相关结果可为预防怪波的危害提供帮助.     

玻色-爱因斯坦凝聚体中的怪波操控

利用Darboux变换法, 解析地研究了玻色-爱因斯坦凝聚体(BEC)中的怪波. 结果表明: 当谱参数等于非线性系数时, BEC中形成一种新型的单洞怪波; 而当谱参数小于非线性系数时, BEC中出现双洞怪波. 进一步地, 怪波的出现位置可通过调节周期性势阱的驱动频率和强度来控制. 此外, 随着原子间相互作用的减小, 怪波的最高幅度也随之降低. 相关结果可为预防怪波的危害提供帮助.     

Exact solutions of the nonlocal Gerdjikov-Ivanov equation

The nonlocal nonlinear Gerdjikov-Ivanov (GI) equation is one of the most important integrable equations, which can be reduced from the third generic deformation of the derivative nonlinear Schrödinger equation. The Darboux transformation is a successful method in solving many nonlocal equations with the help of symbolic computation. As applications, we obtain the bright-dark soliton, breather, rogue wave, kink, W-shaped soliton and periodic solutions of the nonlocal GI equation by constructing its 2n-fold Darboux transformation. These solutions show rich wave structures for selections of different parameters. In all these instances we practically show that these solutions have different properties than the ones for local case.     

Rogue wave solutions of the vector Lakshmanan–Porsezian–Daniel equation

We use a nonrecursive Darboux transformation method to obtain a special hierarchy of rogue wave solutions of the vector Lakshmanan–Porsezian–Daniel equation, which can govern the propagation of ultrashort optical pulses in a long-haul telecommunication fiber. In terms of the exact rational solutions, we demonstrate several interesting rogue wave dynamics such as rogue wave doublets, quartets and sextets. The modulation instability responsible for the excitation of rogue waves from an unstable continuous background in such a complex nonlinear system is also discussed.     

Darboux Transformations,Higher-Order Rational Solitons and Rogue Wave Solutions for a(2+1)-Dimensional Nonlinear Schrdinger Equation

By Taylor expansion of Darboux matrix, a new generalized Darboux transformations(DTs) for a(2 + 1)-dimensional nonlinear Schrdinger(NLS) equation is derived, which can be reduced to two(1 + 1)-dimensional equation:a modified KdV equation and an NLS equation. With the help of symbolic computation, some higher-order rational solutions and rogue wave(RW) solutions are constructed by its(1, N-1)-fold DTs according to determinants. From the dynamic behavior of these rogue waves discussed under some selected parameters, we find that the RWs and solitons are demonstrated some interesting structures including the triangle, pentagon, heptagon profiles, etc. Furthermore, we find that the wave structure can be changed from the higher-order RWs into higher-order rational solitons by modulating the main free parameter. These results may give an explanation and prediction for the corresponding dynamical phenomena in some physically relevant systems.     

Localized waves of the coupled cubic–quintic nonlinear Schrdinger equations in nonlinear optics

We investigate some novel localized waves on the plane wave background in the coupled cubic–quintic nonlinear Schr o¨dinger(CCQNLS) equations through the generalized Darboux transformation(DT). A special vector solution of the Lax pair of the CCQNLS system is elaborately constructed, based on the vector solution, various types of higherorder localized wave solutions of the CCQNLS system are constructed via the generalized DT. These abundant and novel localized waves constructed in the CCQNLS system include higher-order rogue waves, higher-order rogues interacting with multi-soliton or multi-breather separately. The first-and second-order semi-rational localized waves including several free parameters are mainly discussed:(i) the semi-rational solutions degenerate to the first-and second-order vector rogue wave solutions;(ii) hybrid solutions between a first-order rogue wave and a dark or bright soliton, a second-order rogue wave and two dark or bright solitons;(iii) hybrid solutions between a first-order rogue wave and a breather, a second-order rogue wave and two breathers. Some interesting and appealing dynamic properties of these types of localized waves are demonstrated, for example, these nonlinear waves merge with each other markedly by increasing the absolute value of α.These results further uncover some striking dynamic structures in the CCQNLS system.     

Integrable generalization of the associated Camassa–Holm equation

In this paper, we study an integrable generalization of the associated Camassa–Holm equation. The generalized system is shown to be integrable in the sense of Lax pair and the bilinear Bäcklund transformations are presented through the Bell polynomial technique. Meanwhile, its infinite conservation laws are constructed, and conserved densities and fluxes are given in explicit recursion formulas. Furthermore, a Darboux transformation for the system is derived with the help of the gauge transformation between two Lax pairs. As an application, soliton and periodic wave solutions are given through the Darboux transformation.