Vector Rogue Wave Solutions Research Articles

Rogue wave pattern of multi-component derivative nonlinear Schrödinger equations.

This paper studies the multi-component derivative nonlinear Schrödinger (n-DNLS) equations featuring nonzero boundary conditions. Employing the Darboux transformation method, we derive higher-order vector rogue wave solutions for the n-DNLS equations. Specifically, we focus on the distinctive scenario where the (n+1)-order characteristic polynomial possesses an explicit (n+1)-multiple root. Additionally, we provide an in-depth analysis of the asymptotic dynamic behaviors and pattern classification inherent to the higher-order vector rogue wave solution of the n-DNLS equations, mainly when one of the internal arbitrary parameters is extremely large. These patterns are related to the root structures in the generalized Wronskian-Hermite polynomial hierarchies.

Higher-order matrix nonlinear Schrödinger equation with the negative coherent coupling: binary Darboux transformation, vector solitons, breathers and rogue waves

Optical fiber communication plays a crucial role in modern communication. In this work, we focus on the higher-order matrix nonlinear Schrödinger equation with negative coherent coupling in a birefringent fiber. For the slowly varying envelopes of two interacting optical modes, we construct a binary Darboux transformation using the corresponding Lax pair. With vanishing seed solutions and the binary Darboux transformation, we investigate vector degenerate soliton and exponential soliton solutions. By utilizing these soliton solutions, we demonstrate three types of degenerate solitons and double-hump bright solitons. Furthermore, considering non-vanishing seed solutions and applying the binary Darboux transformation, we obtain vector breather solutions, and present the vector single-hump beak-type Akhmediev breather, Kuznetsov-Ma breathers, double-hump beak-type Akhmediev breather, Kuznetsov-Ma breathers, and vector degenerate beak-type breathers. Additionally, we take the limit in the breather solutions and derive vector rogue wave solutions. We illustrate the beak-type rogue waves and bright-dark rogue waves. Humps of these vector double-hump waves can separate into two individual humps. The results obtained in this work may potentially provide valuable insights for experimentally manipulating the separation of two-hump solitons, breathers, and rogue waves in optical fibers.

Vector rogue waves and baseband modulation instability of coupled nonlinear Schrödinger equations for internal gravity waves

In this paper, from the original equations for the two-layer f plane, the third-order coupled nonlinear Schrödinger equations are derived to describe the evolution of two internal gravity waves in the mesoscale atmosphere by using multi-scale analysis and perturbation method. Based on the equation, the vector rogue-wave solutions, which are one of the mechanisms for rainstorm formation, can be obtained through the introduction of two integral preserving transformations. After the modulation instability (MI) analysis, we find the rogue-wave existence is determined by a special MI, namely baseband MI. This may provide a theoretical basis for the prediction of vector rogue waves.

Data-driven parity-time-symmetric vector rogue wave solutions of multi-component nonlinear Schrödinger equation

Rogue waves are a class of nonlinear waves with extreme amplitudes, which usually appear suddenly and disappear without any trace. Recently, the parity-time ()-symmetric vector rogue waves (RWs) of multi-component nonlinear Schrödinger equation (n-NLSE) are usually derived by the methods of integrable systems. In this paper, we utilize the multi-stage physics-informed neural networks (MS-PINNs) algorithm to derive the data-driven symmetric vector RWs solution of coupled NLS system in elliptic and X-shapes domains with nonzero boundary condition. The results of the experiment show that the multi-stage physics-informed neural networks are quite feasible and effective for multi-component nonlinear physical systems in the above domains and boundary conditions.

Exotic Localized Vector Waves in a Two-Component Nonlinear Wave System

A new two-component nonlinear wave system is studied by the generalized perturbation ($$n,N\hbox {-}n$$)-fold Darboux transformation, and various exotic localized vector waves are found. Firstly, the modulational instability is investigated to reveal the mechanism of appearance of rogue waves. Then based on the N-fold Darboux transformation, the generalized perturbation ($$n,N\hbox {-}n$$)-fold Darboux transformation is constructed to solve this two-component nonlinear wave system for the first time. Finally, two types of plane-wave seed solutions are selected to explore the localized vector wave solutions such as vector periodic wave solutions, vector breather solutions, vector rogue wave solutions and vector interaction solutions. It is found that there are both localized bright–dark vector waves and bright–bright vector waves in this system, which have not been reported before.

Breather waves, high-order rogue waves and their dynamics in the coupled nonlinear Schrödinger equations with alternate signs of nonlinearities

We consider the analytic vector breather and high-order rogue wave solutions for the coupled nonlinear Schrödinger (NLS) equations with alternate signs of nonlinearities via Darboux dressing transformation. By adjusting spectral parameters, we indicate that the breather wave solutions contain temporally periodic nonlinear waves and spatially periodic nonlinear waves, respectively. The vector rogue wave solutions include the bright one-peak-two-valleys rogue wave and the bright rogue wave without valleys. Additionally, we successfully show different types of the distributions for the second-order vector rogue waves. The existence condition for the rogue waves is discussed. For the coupled NLS equations with alternate signs of nonlinearities, the rogue wave solutions exist if the baseband modulation instability (MI) is present.

Localized waves of the coupled cubic–quintic nonlinear Schrödinger equations in nonlinear optics**Project supported by the Global Change Research Program of China (Grant No. 2015CB953904), the National Natural Science Foundation of China (Grant Nos. 11675054 and 11435005), the Shanghai Collaborative Innovation Center of Trustworthy Software for Internet of Things (Grant No. ZF1213), and the Natural Science Foundation of Hebei Province, China

We investigate some novel localized waves on the plane wave background in the coupled cubic–quintic nonlinear Schrödinger (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.

Interactions of localized wave structures and dynamics in the defocusing coupled nonlinear Schrödinger equations.

We investigate the defocusing coupled nonlinear Schrödinger equations from a 3×3 Lax pair. The Darboux transformations with the nonzero plane-wave solutions are presented to derive the newly localized wave solutions including dark-dark and bright-dark solitons, breather-breather solutions, and different types of new vector rogue wave solutions, as well as interactions between distinct types of localized wave solutions. Moreover, we analyze these solutions by means of parameters modulation. Finally, the perturbed wave propagations of some obtained solutions are explored by means of systematic simulations, which demonstrates that nearly stable and strongly unstable solutions. Our research results could constitute a significant contribution to explore the distinct nonlinear waves (e.g., dark solitons, breather solutions, and rogue wave solutions) dynamics of the coupled system in related fields such as nonlinear optics, plasma physics, oceanography, and Bose-Einstein condensates.

On the characterization of vector rogue waves in two-dimensional two coupled nonlinear Schrödinger equations with distributed coefficients

We construct vector rogue wave solutions of the two-dimensional two coupled nonlinear Schr\"{o}dinger equations with distributed coefficients, namely diffraction, nonlinearity and gain parameters through similarity transformation technique. We transform the two-dimensional two coupled variable coefficients nonlinear Schr\"{o}dinger equations into Manakov equation with a constraint that connects diffraction and gain parameters with nonlinearity parameter. We investigate the characteristics of the constructed vector rogue wave solutions with four different forms of diffraction parameters. We report some interesting patterns that occur in the rogue wave structures. Further, we construct vector dark rogue wave solutions of the two-dimensional two coupled nonlinear Schr\"{o}dinger equations with distributed coefficients and report some novel characteristics that we observe in the vector dark rogue wave solutions.

Vector rogue waves in the mixed coupled nonlinear Schrödinger equations

In this paper, via the generalized Darboux transformation we derive the reduced and non-reduced vector rogue wave solutions of the focusing-defocusing mixed coupled nonlinear Schrodinger equations. The dynamics of reduced vector rogue waves is the same as that for the known scalar ones. The non-reduced solutions can exhibit both the one-peak-two-valleys structure with one peak and two valleys lying in a straight line, and the two-peaks-two-valleys structure with two peaks and two valleys located at the four vertices of a parallelogram. We also find that the amplitude of the non-reduced vector rogue wave is not three times as that of the exciting plane wave, and that the coalescence of multiple fundamental rogue waves does not generate larger-amplitude rogue waves. In addition, we discuss the relationship of the free parameters in the solutions with the positions and relative distances of rogue waves in the xt-plane.

Rogue wave solutions for the coupled cubic–quintic nonlinear Schrödinger equations with variable coefficients

In this paper, by the Darboux transformation, we get the vector rogue wave solutions of the coupled cubic–quintic nonlinear Schrödinger equations with variable coefficients, which come from twin-core nonlinear optical fibers and waveguides, describing the effects of quintic nonlinearity on the ultrashort optical pulse propagation in the non-Kerr media. The first-order vector rogue wave solutions obtained in this paper can admit certain different patterns. Dynamical features of the rogue wave solutions are graphically investigated. These results might be of some value for the ultrashort optical pulse propagation in the non-Kerr media.