Higher-order Rogue Waves Research Articles

Rogue wave patterns of two-component nonlinear Schrödinger equation coupled to the Boussinesq equation

This paper examines rogue wave patterns in the two-component nonlinear Schrödinger equation coupled with the Boussinesq equation. Using the Kadomtsev–Petviashvili (KP) hierarchy reduction method, we derive rogue wave solutions for the case where a sextic equation has a pair of complex conjugate roots with multiplicity two. By selecting a simpler differential operator, the solutions become clearer and more concise, and they are expressed in terms of Schur polynomials. We also investigate the stability of these solutions through numerical simulations. As the single internal parameter of high-order rogue wave solutions increases, the rogue waves form distinct geometric patterns, with fundamental rogue waves arranged in structures such as double triangles, diamonds, rectangles, pentagons, and heptagons, and a possible lower-order rogue wave at the center. The positions of these outer rogue waves are closely related to the non-zero roots of the Okamoto polynomial hierarchy. The positions of these outer rogue waves are closely related to the non-zero roots of the Okamoto polynomial hierarchy. These patterns are analytically determined by the root structure of the Okamoto polynomial hierarchy through translation, dilation, stretch, shear and rotation. The comparison between the analytical predictions and the actual solutions of these rogue wave patterns shows remarkable agreement.

Pattern dynamics of higher-order rogue waves in the nonlinear Schrödinger–Boussinesq equation

Pattern dynamics of higher-order rogue waves in the nonlinear Schrödinger–Boussinesq equation

Breather–breather interactions, rational and semi-rational solutions of an integrable spin system in optical fibers

This paper investigates an integrable spin system (or Heisenberg ferromagnet-type equation or Nurshuak-IIA equation) which is an integrable generalization of the Heisenberg ferromagnet equation that raises from magneto-optical devices. The N-fold Darboux transformation of the Nurshuak-IIA equation will be constructed. With the selection of several parameters, a variety of the solutions will be produced, including breather–breather interactions, higher-order rogue waves and semi-rational solutions.

Rogue-wave structures for a generalized (3+1)-dimensional nonlinear wave equation in liquid with gas bubbles

This study explores the behavior of higher-order rogue waves within a (3+1)-dimensional generalized nonlinear wave equation in liquid-containing gas bubbles. It creates the investigated equation’s Hirota D-operator bilinear form. We employ a generalized formula with real parameters to obtain the rogue waves up to the third order using the direct symbolic technique. The analysis reveals that the second and third-order rogue solutions produce two and three-waves, respectively. To gain deeper insights, we use the Cole-Hopf transformation on the transformed variables ξ and η to produce a bilinear equation. Using the system software Mathematica, the dynamic analysis presents the graphics for the obtained solutions in transformed ξ, η, and original spatial-temporal coordinates x, y, z, t. These visualizations reveal rogue waves’ intricate structure and evolution, capturing their localized interactions and significant presence within nonlinear systems. We demonstrate that rogue waves, characterized by their substantial height and sudden appearance, are prevalent in various nonlinear events. The equation examined in this study offers valuable insights into the evolution of longer waves with smaller amplitudes, which is particularly relevant in fields such as fluid dynamics, dispersive media, and plasmas. The implications of this research extend across multiple scientific domains, including fiber optics, oceanography, dusty plasma, and nonlinear systems, where understanding the behavior of rogue waves is crucial for both theoretical and practical applications.

Vector rogue waves in spin-1 Bose–Einstein condensates with spin–orbit coupling

Abstract We analytically and numerically study three-component rogue waves (RWs) in spin-1 Bose–Einstein condensates with Raman-induced spin–orbit coupling (SOC). Using the multiscale perturbative method, we obtain approximate analytical solutions for RWs with positive and negative effective masses, determined by the effective dispersion of the system. The solutions include RWs with smooth and striped shapes, as well as higher-order RWs. The analytical solutions demonstrate that the RWs in the three components of the system exhibit different velocities and their maximum peaks appear at the same spatiotemporal position, which is caused by SOC and interactions. The accuracy of the approximate analytical solutions is corroborated by comparison with direct numerical simulations of the underlying system. Additionally, we systematically explore existence domains for the RWs determined by the baseband modulational instability (BMI). Numerical simulations corroborate that, under the action of BMI, plane waves with random initial perturbations excite RWs, as predicted by the approximate analytical solutions.

Modulation instability and rogue waves for two and three dimensional nonlinear Klein-Gordon equation.

We perform the modulation instability analysis of the 2D and 3D nonlinear Klein-Gordon equation. The instability region depends on dispersion and wavenumbers of the plane wave. The N-breathers of the nonlinear Klein-Gordon equation are constructed directly from its 2N-solitons obtained in history. The regularity conditions of breathers are established. The dynamic behaviors of breathers of the 2D nonlinear Klein-Gordon equation are consistent with modulation instability analysis. Furthermore, by means of the bilinear method together with improved long-wave limit technique, we obtain general high order rogue waves of the 2D and 3D nonlinear Klein-Gordon equation. In particular, the first- and second-order rogue waves and lumps of the 2D nonlinear Klein-Gordon equation are investigated by using their explicit expressions. We find that their dynamic behaviors are similar to the nonlinear Schrödinger equation. Finally, the first-order rational solutions are illustrated for the 3D nonlinear Klein-Gordon equation. It is demonstrated that the rogue waves of the 2D and 3D nonlinear Klein-Gordon equation always exist by choosing dispersion and wavenumber of plane waves.

A high-order rogue wave generated by collision in three-component Bose–Einstein condensates

The generation of high-order rogue waves (RWs) without the exact solution is an intriguing subject that has not yet been fully explored, especially for the third-order RWs. In this paper, we investigate the collision dynamics in a three-component Bose–Einstein condensate (BEC), and propose a scheme capable of generating such high-order RWs, such as second- and third-order RWs. The results show that the peaks of the three first-order RWs coincide in time and space when the third-order RWs are excited during the collision. Furthermore, by controlling the offsets of initial wavepackets and intraspecific interaction coefficients within the BEC, the collision behavior of RWs can be precisely manipulated. Compared to two-component BECs, these three-component collisions exhibit more diverse structures for exciting RW phenomena.

Mixed localized waves and their interaction structures for a spatial discrete Hirota equation

Mixed localized wave solutions and interactions are of great significance in nonlinear physical systems. This paper aims to investigating the generalized (m,N-m)-fold Darboux transformation and the mixed localized wave solutions of a spatial discrete Hirota equation. First, we construct the generalized (m,N-m)-fold Darboux transformation for the spatial discrete Hirota equation, which can produce the interactions between the breathers, degenerate breathers and rogue waves. For the Darboux transformation formula, we discuss the above order-1,2,3 localized wave solutions, as well as their dynamics by choosing the number of m = 1. We plot some specific examples such as the spatial (time)-periodic breather, second-order and third-order degenerate breathers and higher-order rogue waves with novel patterns. Furthermore, when m > 1, we give several kinds of mixed interaction solutions between the first-order rogue waves and first (second)-order (degenerate) breathers, between the first-order breather and second-order degenerate breathers, between second-order rogue waves and first-order breathers. At last, we also sum up the various mathematical features of the degenerate breathers and the mixed localized wave solutions.

Coupled Hirota system: higher-order rogue waves and multi-solitons structures

Coupled Hirota system: higher-order rogue waves and multi-solitons structures

Controllable nonautonomous localized waves and dynamics for a quasi-1D Gross-Pitaevskii equation in Bose-Einstein condensations with attractive interaction.

This paper investigates dynamical behaviors and controllability of some nonautonomous localized waves based on the Gross-Pitaevskii equation with attractive interatomic interactions. Our approach is a relation constructed between the Gross-Pitaevskii equation and the standard nonlinear Schrödinger equation through a new self-similarity transformation which is to convert the exact solutions of the latter to the former's. Subsequently, one can obtain the nonautonomous breather solutions and higher-order rogue wave solutions of the Gross-Pitaevskii equation. It has been shown that the nonautonomous localized waves can be controlled by the parameters within the self-similarity transformation, rather than relying solely on the nonlinear intensity, spectral parameters, and external potential. The control mechanism can induce an unusual number of loosely bound higher-order rogue waves. The asymptotic analysis of unusual loosely bound rogue waves shows that their essence is energy transfer among rogue waves. Numerical simulations test the dynamical stability of obtained localized wave solutions, which indicate that modifying the parameters in the self-similarity transformation can improve the stability of unstable localized waves and prolong their lifespan. We numerically confirm that the rogue wave controlled by the self-similarity transformation can be reproduced from a chaotic initial background field, hence anticipating the feasibility of its experimental observation, and propose an experimental method for observing these phenomena in Bose-Einstein condensates. The method presented in this paper can help to induce and observe new stable localized waves in some physical systems.

High-order rogue wave and mixed interaction patterns for the three-component Gross-Pitaevskii equationsin F=1 spinor Bose-Einstein condensates.

Under investigation are the three-component Gross-Pitaevskii equationsin F=1 spinor Bose-Einstein condensates. Various localized waves' generation mechanisms have been derived from plane wave solutions using modulation instability. The perturbed continuous waves produce a large number of rogue wave structures through the split-step Fourier numerical method. Based on the known Lax pair, we construct the generalized iterative (n,N-n)-fold Darboux transformation to generate various high-order solutions, including the bright-dark-bright structure of rogue waves, periodic waves, and their mixed interaction structures. Numerical simulations show that rogue waves with a two-peaked structure have robust noise resistance and stable dynamical behavior. The asymptotic states of high-order rogue waves as the parameter approaches infinity are also predicted using the large parameter asymptotic technique. In addition, the position of these localized wave patterns can be controlled by some special parameters. These results may help us understand the dynamic behavior of spinor condensates for the mean-field approximation.

Adaptive sampling physics-informed neural network method for high-order rogue waves and parameters discovery of the (2 + 1)-dimensional CHKP equation.

Rogue waves are important physical phenomena, which have wide applications in nonlinear optics, hydrodynamics, Bose-Einstein condensates, and oceanic and atmospheric dynamics. We find that when using the original PINNs to study rogue waves of high dimensional PDEs, the prediction performance will become very poor, especially for high-order rogue waves due to that the randomness of selection of sample points makes insufficient use of the physical information describing the local sharp regions of rogue waves. In this paper, we propose an adaptive sampling physics-informed neural network method (ASPINN), which renders the points in local sharp regions to be selected sufficiently by a new adaptive search algorithm to lead to a prefect prediction performance. To valid the performance of our method, the (2+1)-dimensional CHKP equation is taken as an illustrative example. Experimental results reveal that the original PINNs can hardly be able to predict dynamical behaviors of the high-order rogue waves for the CHKP equation, but the ASPINN method can not only predict dynamical behaviors of these high-order rogue waves, but also greatly improve the prediction efficiency and accuracy to four orders of magnitude. Then, the data-driven inverse problem for the CHKP equation with different levels of corrupted noise is studied to show that the ASPINN method has good robustness. Moreover, some main factors affecting the neural network performance are discussed in detail, including the size of training data, the number of layers of the neural network, and the number of neurons per layer.

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.

Large-time lump patterns of Kadomtsev-Petviashvili I equation in a plasma analyzed via vector one-constraint method

In plasma physics, the Kadomtsev–Petviashvili I (KPI) equation is a fundamental model for investigating the evolution characteristics of nonlinear waves. For the KPI equation, the constraint method is an effective tool for generating solitonic or rational solutions from the solutions of lower-dimensional integrable systems. In this work, various nonsingular, rational lump solutions of the KPI equation are constructed by employing the vector one-constraint method and the generalized Darboux transformation of the (1 + 1)-dimensional vector Ablowitz–Kaup–Newell–Segur system. Furthermore, we investigate the large-time asymptotic behavior of high-order lumps in detail and discover distinct types of patterns. These lump patterns correspond to the high-order rogue wave patterns of the (1 + 1)-dimensional vector integrable equation and are associated with root structures of generalized Wronskian–Hermite polynomials.

Higher-order hybrid rogue wave and breather interaction dynamics for the AB system in two-layer fluids

Rogue waves and breathers emerge as significant solitons, result from ubiquitous nonlinear effects of dynamics in the natural world, science and engineering. In this paper, we study the AB system, a model derived from a two-layer fluid that is widely used to investigate nonlinear fluid dynamics. We use the Darboux transformation to construct analytical solutions for higher-order hybrid rogue waves and breathers, which are expressed by symmetric algebraic matrices. We use these solutions and the spectrum parameters to explore the complex interaction dynamics of the hybrid rogue waves and breathers. We find that the interactions cause significant changes in the phases and shapes. Moreover, we discover an interesting phenomenon: the collisions between the rogue waves and breathers are semi-elastic, meaning that only the phases of the rogue waves change, while the speeds, shapes and phases of the breathers remain the same, before and after the collisions. This study provides new insights into the dynamics of the AB system, and helps us understand nonlinear phenomena in various two-layer fluid systems.

Multi-breathers and higher-order rogue waves on the periodic background in a fourth-order integrable nonlinear Schrödinger equation

In this paper, we present a systematic formulation of multi-breathers and higher-order rogue wave solutions of a fourth-order nonlinear Schrödinger equation on the periodic background. First of all, we compute a complete family of elliptic solution of this higher-order equation, which can degenerate into two particular cases, i.e., the dnoidal and cnoidal solutions. By using the modified squared wavefunction approach, we solve the spectral problem on the elliptic function background. Then, we derive multi-breather solutions in terms of the theta functions, particular examples of which are the Kuznetsov-Ma breather and the Akhmediev breather. Furthermore, taking the limit of the breather solutions at branch points, we construct higher-order rogue wave solutions by employing a generalized Darboux transformation technique. On the periodic background, we present the first-order, second-order and second-second-order rogue waves. With aid of the theta functions, we explicitly characterize the resulting breathers and rogue waves, and demonstrate their dynamic behaviors by illustrative examples. Finally, we discuss how the parameter of the higher-order effects affects the breathers and rogue waves.

The improved backward compatible physics-informed neural networks for reducing error accumulation and applications in data-driven higher-order rogue waves.

Due to the dynamic characteristics of instantaneity and steepness, employing domain decomposition techniques for simulating rogue wave solutions is highly appropriate. Wherein, the backward compatible physics-informed neural network (bc-PINN) is a temporally sequential scheme to solve PDEs over successive time segments while satisfying all previously obtained solutions. In this work, we propose improvements to the original bc-PINN algorithm in two aspects based on the characteristics of error propagation. One is to modify the loss term for ensuring backward compatibility by selecting the earliest learned solution for each sub-domain as pseudo-reference solution. The other is to adopt the concatenation of solutions obtained from individual subnetworks as the final form of the predicted solution. The improved backward compatible PINN (Ibc-PINN) is applied to study data-driven higher-order rogue waves for the nonlinear Schrödinger (NLS) equation and the AB system to demonstrate the effectiveness and advantages. Transfer learning and initial condition guided learning (ICGL) techniques are also utilized to accelerate the training. Moreover, the error analysis is conducted on each sub-domain, and it turns out that the slowdown of Ibc-PINN in error accumulation speed can yield greater advantages in accuracy. In short, numerical results fully indicate that Ibc-PINN significantly outperforms bc-PINN in terms of accuracy and stability without sacrificing efficiency.

Dark Localized Waves in Shallow Waters: Analysis within an Extended Boussinesq System

We study dark localized waves within a nonlinear system based on the Boussinesq approximation, describing the dynamics of shallow water waves. Employing symbolic calculus, we apply the Hirota bilinear method to transform an extended Boussinesq system into a bilinear form, and then use the multiple rogue wave method to obtain its dark rational solutions. Exploring the first- and second-order dark solutions, we examine the conditions under which these localized solutions exist and their spatiotemporal distributions. Through the selection of various parameters and by utilizing different visualization techniques (intensity distributions and contour plots), we explore the dynamical properties of dark solutions found: in particular, the first- and second-order dark rogue waves. We also explore the methods of their control. The findings presented here not only deepen the understanding of physical phenomena described by the (1+1)-dimensional Boussinesq equation, but also expand avenues for further research. Our method can be extended to other nonlinear systems, to conceivably obtain higher-order dark rogue waves.

Rogue waves and their patterns for the coupled Fokas–Lenells equations

In this work, we explore the rogue wave patterns in the coupled Fokas–Lenells equation by using the Darboux transformation. We demonstrate that when one of the internal parameters is large enough, the general high-order rogue wave solutions generated at a branch point of multiplicity three can be decomposed into some first-order outer rogue waves and a lower-order inner rogue wave. Remarkably, the positions and the orders of these outer and inner rogue waves are intimately related to Okamoto polynomial hierarchies.

Robust inverse scattering analysis of discrete high-order nonlinear Schrödinger equation

Abstract In this study, we present the rigorous theory of the robust inverse scattering method for the discrete high-order nonlinear Schrödinger (HNLS) equation with a nonzero boundary condition (NZBC). Using the direct scattering problem, we deduce the analyticity, symmetries, and asymptotic behaviors of the Jost solutions and scattering matrix. We also formulate the inverse scattering problem using the matrix Riemann–Hilbert problem (RHP). Furthermore, utilizing the loop group theory, we construct the multi-fold Darboux transformation (DT) within the framework of the robust inverse scattering transform. Additionally, we develop the corresponding Bäcklund transformation (BT) to obtain the multi-fold lattice soliton solutions. To derive the high-order rational solutions, we further construct the high-order DT. Finally, we theoretically and graphically analyze these solutions, which exhibit lattice breather waves, W-shape lattice solitons, high-order lattice rogue waves (RW), and their interactions.