Rogue Wave Solutions Research Articles

Non-autonomous exact solutions and dynamic behaviors for the variable coefficient nonlinear Schrödinger equation with external potential

Abstract The nonlinear Schrödinger(NLS) equation represents a nonlinear dynamical system, which is usually used to describe nonlinear waves in deep water, self-focusing of intense lasers contained in electrolytes, and so on. The exact solutions of the variable coefficient nonlinear Schrödinger equation with external potential are considered. The variable coefficient nonlinear Schrödinger equation is transformed into a constant coefficient nonlinear equation by using the similarity transformation method, the exact solutions of the constant coefficient nonlinear equation are generated by using the homogeneous balance method. We obtain one-soliton, two-soliton, kink type soliton, bright soliton, parabolic soliton and rogue wave solutions. A ‘stepped’ type soliton solution is obtained in this paper, which is a novel type solution and different from the solutions of most nonlinear Schrödinger equations. Some special dynamic behaviors of solitons of the variable coefficient nonlinear Schrödinger equation with external potential are obtained via selecting some free functions. We found that the numerical simulation result is consistent with the exact solution through illustrating the time evolution of bright soliton solution.

PINN for solving forward and inverse problems involving integrable two-dimensional nonlocal equations

In this paper, the physics informed neural network (PINN) deep learning method is applied to solve two-dimensional nonlocal equations, including the partial reverse space y-nonlocal Mel’nikov equation, the partial reverse space-time nonlocal Mel’nikov equation and the nonlocal two-dimensional nonlinear Schrödinger (NLS) equation. By the PINN method, we successfully derive a data-driven two soliton solution, lump solution and rogue wave solution. Numerical simulation results indicate that the error range between the data-driven solution and the exact solution is relatively small, which verifies the effectiveness of the PINN deep learning method for solving high dimensional nonlocal equations. Moreover, the parameter discovery of the partial reverse space-time nonlocal Mel’nikov equation is analysed in terms of its soliton solution for the first time.

The breather, breather-positon, rogue wave for the reverse space–time nonlocal short pulse equation in nonzero background

In this paper, by using the Darboux transformation (DT), two types of breather solutions for the reverse space–time (RST) nonlocal short pulse equation are constructed in nonzero background: bounded and unbounded breather solutions. The degenerate DT is obtained by taking the limit of eigenvalues and performing a higher-order Taylor expansion. Then the N order breather-positon solutions are generated through degenerate DT. Some properties of the breather-positon solutions are discussed. Furthermore, rogue wave solutions are derived through the degeneration of breather-positon solutions.

Rogue wave patterns in the nonlocal nonlinear Schrödinger equation

This paper investigates rogue wave patterns in the nonlocal nonlinear Schrödinger (NLS) equation. Initially, employing the Kadomtsev–Petviashvili reduction method, rogue wave solutions of the nonlocal NLS equation, whose τ function is a 2×2 block matrix, are simplified. Afterward, utilizing the asymptotic analysis approach, we investigate the rogue wave patterns when two free parameters a2m1+1 and b2m2+1 are considerably large and fulfill the condition |a2m1+1|2/(2m1+1)=O(|b2m2+1|1/(2m2+1)). Our findings reveal that under these conditions, rogue wave solutions of the nonlocal NLS equation exhibit novel patterns, which consist of three regions, which are the outer region, the middle region and the inner region. In the outer and middle regions, only single rogue waves with singularities may occur, and their locations are characterized by roots of two polynomials from the Yablonskii–Vorob'ev polynomial hierarchies. In the inner region, a possible lower order rogue wave may appear, which can be singular or regular, depending on the values of m1,m2, the sizes of τ function, and certain free parameters. Finally, the numerical results indicate that the predicted outcomes are in close alignment with real rogue waves.

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.

Exploring N-soliton solutions, multiple rogue wave and the linear superposition principle to the generalized hirota satsuma-ito equation

In this work, numerous rogue wave solution strategy (MRWSS) agreeing to the Hirota bilinear hypothesis is utilized. The said strategy to build different soliton wave solutions for the generalized Hirota-Satsuma-Ito (GHSI) condition is considered. These numerous soliton wave solutions incorporate first-order, second-order, third-order, and fourth-order waves solutions and so on, which are all depending on the starting theory of chosen capacities. For the lump solution with one most extreme or least esteem, the properties of the Hessian lattice are examined. Besides, the arrangements are contained the lump-two kink arrangements, cross-kink wave arrangements, periodic wave arrangements, and periodic-kink wave arrangements. The various classes of arrangements are gotten. We make utilize of the direct superposition procedure to find out N-soliton wave arrangements to the said condition. The appropriateness and viability of the obtained arrangements is detailed through the reenactment comes about within the shape of 3-D, density, and 2-D charts. The gotten comes about in this work are anticipated to open modern viewpoints for the traveling wave theory.

Rogue wave patterns associated with Adler–Moser polynomials featuring multiple roots in the nonlinear Schrödinger equation

AbstractIn this work, we analyze the asymptotic behaviors of high‐order rogue wave solutions with multiple large parameters and discover novel rogue wave patterns, including modified claw‐like, one triple root (OTR)‐type, modified OTR‐type, two triple roots (TTR)‐type, semimodified TTR‐type, and modified TTR‐type patterns. A correlation is established between these rogue wave patterns and the root structures of the Adler–Moser polynomials with multiple roots. At the positions in the ‐plane corresponding to simple roots of the Adler–Moser polynomials, these high‐order rogue wave patterns asymptotically approach first‐order rogue waves. At the positions in the ‐plane corresponding to multiple roots of the Adler–Moser polynomials, these rogue wave patterns asymptotically tend toward lower‐order fundamental rogue waves, dispersed first‐order rogue waves, or mixed structures of these rogue waves. These structures are related to the root structures of special Adler–Moser polynomials with new free parameters, such as the Yablonskii–Vorob'ev polynomial hierarchy, among others. Notably, the positions of the fundamental lower‐order rogue waves or mixed structures in these rogue wave patterns can be controlled freely under specific conditions.

Study of Fifth-Order Nonlinear Caudrey–Dodd–Gibbon Model for Multiwave, M-Shaped Solitons, Lumps, Generalized Breathers, Manifold Periodic, Rogue Wave and Kink Wave Interactions

Numerous novel wave structures, such as lump, lump with one-kink, lump with two-kink soliton, generalized breather, Ma breather, Kuznetsov-Ma breather, Akhmediev breather, manifold periodic, and rogue wave solutions to [Formula: see text]-dimensional fifth-order nonlinear Caudrey–Dodd–Gibbon (FNCDG) model via symbolic computation are studied based on the ansatz function scheme. Studying the lump-type solution involves selecting the function as the generic quadratic polynomial function. The lump with one- and two-kink solutions is created by mixing a quadratic function with one or two exponential functions. By adding hyperbolic function with a quadratic function, the rogue wave solutions are manipulated. In addition, the multiwave, [Formula: see text]-shaped rational solitons and their interactions with kink waves are analytically evaluated. Furthermore, different 3D, 2D, and contour profiles are created in different dimensions by changing the parameter values.

Stochastic soliton and rogue wave solutions of the nonlinear Schrödinger equation with white Gaussian noise

In this letter, we propose a novel integrable nonlinear Schrödinger equation and its Lax pair, influenced by Brownian motion and white Gaussian noise. We aim to construct and solve new integrable systems affected by the white Gaussian noise. Utilising the classical and generalised Darboux transformations, the stochastic soliton solutions and the stochastic rogue wave solutions of this novel integrable nonlinear Schrödinger equation are obtained and expressed in determinant form. Studies of stochastic soliton and rogue wave solutions of the NLS equation are essential for complex physical and mathematical phenomena where nonlinear interactions and randomness play crucial roles.

Modulational instability, generation, and evolution of rogue waves in the generalized fractional nonlinear Schrödinger equations with power-law nonlinearity and rational potentials.

In this paper, we investigate several properties of the modulational instability (MI) and rogue waves (RWs) within the framework of the generalized fractional nonlinear Schrödinger (FNLS) equations with rational potentials. We derive the dispersion relation for a continuous wave (CW), elucidating the relationship between the wavenumber and the instability growth rate of the CW solution in the absence of potentials. This relationship is primarily influenced by the power parameter σ, the Lévy index α, and the nonlinear coefficient g. Our theoretical findings are corroborated by numerical simulations, which demonstrate that MI occurs in the focusing context. Furthermore, we study the RW generations in both cubic and quintic FNLS equations with two types of time-dependent rational potentials, which make both cubic and quintic NLS equations support the exact RW solutions. Specifically, we show that the introduction of these two potentials allows for the excitations of controllable RWs in the defocusing regime. When these two potentials become the time-independent cases such that the stable W-shaped solitons with non-zero backgrounds are generated in these cubic and quintic FNLS equations. Moreover, we consider the excitations of higher-order RWs and investigate the conditions necessary for their generations. Our analysis reveals the intricate interplay between the system parameters and the potential configurations, offering insights into the mechanisms that facilitate the emergence of higher-order RWs. Finally, we find the separated controllable multi-RWs in the defocusing cubic FNLS equation with time-dependent multi-potentials. This comprehensive study not only enhances our understanding of MI and RWs in the fractional nonlinear wave systems, but also paves the way for future research in related nonlinear wave phenomena.

Rogue waves on the periodic background for a higher-order nonlinear Schrödinger–Maxwell–Bloch system

In this paper, we construct the rogue wave solutions on the background of the Jacobian elliptic functions for a higher-order nonlinear Schrödinger–Maxwell–Bloch system. The Jacobian elliptic function traveling wave solutions as the seed solutions are considered. Through the approach of the nonlinearization of the Lax pair and Darboux transformation method, the rogue waves and the line rogue waves on the Jacobian elliptic functions dn and cn background are obtained, respectively.

Rogue waves of the Nizhnik-Novikov-Veselov equation via self-mapping transformation

This paper introduces a new two-dimensional self-mapping transformation applied to the Nizhnik-Novikov-Veselov equation, resulting in the generation of numerous rogue wave solutions. We discover that temporal-localized and spatiotemporal-localized two-dimensional rogue waves respectively. Notably, these rogue waves emerge from a zero background and subsequently exhibit both algebraic and exponential decay patterns. The proposed technique offers a potential tool for constructing rogue-like waves within (2+1)-dimensional nonlinear wave frameworks. The findings presented here serve as a robust mathematical foundation for advancing both theoretical understanding and practical applications of rogue waves.

Retrieval of lump, breather, interactions, and rogue wave solutions to the fractional complex paraxial wave dynamical model with sensitivity analysis

In this research, the Hirota bilinear method and the modified Sardar sub-equation (MSSE) techniques are used to investigate the generation and detection of soliton structures in the fractional complex paraxial wave dynamical (FPWD) model together with Kerr media. By employing the aforementioned techniques, we derive lump and different exact solitary wave solutions for the selected model, which has not been documented in previous literature. We manifested some novel lump soliton solutions, including the homoclinic breather wave, periodic cross rational wave, the M-shaped interaction with rogue and kink waves, the M-shaped rational solution, the M-shaped rational solution with one and two kink waves, and multi-wave solutions. Furthermore, for intellectual curiosity, we also amalgamated the rich spectrum of soliton solutions such as W-shape, periodic, dark, bright, combo, rational, exponential, mixed trigonometric, and hyperbolic soliton wave solutions inherent in the FPWD equation. We also undertake sensitivity analysis to examine the resilience of the selected model in the face of variations in initial circumstances and parameters, which provides insights into the system’s sensitivity to perturbations. Furthermore, we investigate the ramifications of these findings for a variety of physical systems, including optics, fluid dynamics, and plasma physics. These findings are to gain a better knowledge of nonlinear wave phenomena and fresh insights into the dynamics of complex systems by combining the Hirota bilinear technique and the MSSE method.

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.

Multiple rogue wave, double-periodic soliton and breather wave solutions for a generalized breaking soliton system in (3 + 1)-dimensions

We focused on solitonic phenomena in wave propagation which was extracted from a generalized breaking soliton system in (3 + 1)-dimensions. The model describes the interaction phenomena between Riemann wave and long wave via two space variable in nonlinear media. Abundant double-periodic soliton, breather wave and the multiple rogue wave solutions to a generalized breaking soliton system by the Hirota bilinear form and a mixture of exponentials and trigonometric functions are presented. Periodic-soliton, breather wave and periodic are studied with the usage of symbolic computation. In addition, the symbolic computation and the applied methods for governing model are investigated. Through three-dimensional graph, density graph, and two-dimensional design using Maple, the physical features of double-periodic soliton and breather wave solutions are explained all right. The findings demonstrate the investigated model’s broad variety of explicit solutions. All outcomes in this work are necessary to understand the physical meaning and behavior of the explored results and shed light on the significance of the investigation of several nonlinear wave phenomena in sciences and engineering.

Modulation instability, state transitions and dynamics of multi-peak rogue wave in a higher-order coupled nonlinear Schrödinger equation

The coupled nonlinear Schrödinger equation provides an effective description of the propagation of four optical fields in the fibers. The present work explores the dynamics of rogue waves that arise in a coupled nonlinear Schrödinger equation of higher-order that is acquired using the Ablowitz-Kaup-Newell-Segur method. The generalized (m,N−m)-fold Darboux transformation is used to determine the exact rogue wave solutions, which show multiple peaks and depressions. In addition, a detailed analysis is conducted of the modulation instability of three different kinds of plane-wave solutions. The results indicate that the coefficients of the second and third derivative terms, rather than just the coefficients of the third derivative term, have an impact on modulation instability. Lastly, the transition between rogue waves and solitons under the influence of higher-order dispersion effects is elucidated, with explicit soliton solutions provided within the modulation stability region.

Patterns of rogue waves in the sharp-line Maxwell–Bloch system

The Maxwell–Bloch system describes light-matter interactions in a semi-infinitely long one dimensional two-level optical medium. Rogue wave patterns in the Maxwell–Bloch system under sharp-line limit are analytically studied. It is shown that when single internal parameter in bilinear expressions of rogue waves gets large, these waves would exhibit clear geometric patterns, which comprise fundamental (Peregrine) rogue waves arranged in shapes such as triangle, pentagon, heptagon and nonagon structures, with a possible lower-order rogue wave at the center. These rogue wave patterns are analytically determined from the root structure of the Yablonskii–Vorob’ev polynomial hierarchy through dilation, rotation, stretch and shear. It is also shown that when multiple internal parameters in the rogue wave solutions get large, new rogue wave patterns would arise, including heart-shaped structures, fan-shaped structures, and many others. Analytically, these patterns are determined by the root structure of the Adler–Moser polynomials through a linear transformation. Comparison between analytical predictions of these rogue patterns and true solutions shows excellent agreement.

Rogue waves in a reverse space nonlocal nonlinear Schrödinger equation

Rogue waves in a reverse space nonlocal nonlinear Schrödinger (NLS) equation with real and parity-symmetric nonlinearity-induced potential are considered. This equation has clear physical meanings since it can be derived from the Manakov system with a special reduction. The N-fold Darboux transformation and its generalized form for the nonlocal NLS equation are constructed. As an application, the multiparametric Nth-order rogue wave solution in terms of Schur polynomials for the nonlocal NLS equation with focusing case is derived by the limit technique. The significant differences of rogue wave dynamics between the nonlocal NLS equation and its usual (local) counterpart are illustrated through two types of specific rogue wave solutions. Unlike the eye-shaped (Peregrine type) rogue waves, the rogue wave doublets which involve an eye-shaped rogue wave and a dark/four-petaled rogue wave merging or separating with each other, and the rogue wave sextets that are characterized by the superpositions of three eye-shaped rogue waves and three dark/four-petaled rogue waves with fundamental, triangular and quadrilateral patterns are shown. Moreover, some wave characteristics including the difference between the light intensity and the plane-wave background, and the pulse energy of the rogue wave doublets are discussed.

Multiple rogue wave solutions for a modified (2 + 1)-dimensional nonlinear evolution equation

Multiple rogue wave solutions for a modified (2 + 1)-dimensional nonlinear evolution equation

Extreme Superposition: High-Order Fundamental Rogue Waves in the Far-Field Regime

This Memoir concerns fundamental rogue-wave solutions of the focusing nonlinear Schrödinger equation in the limit that the order of the rogue wave is large and the independent variables ( x , t ) (x,t) are proportional to the order (the far-field limit). We first formulate a Riemann-Hilbert representation of these solutions that allows the order to vary continuously rather than by integer increments. The intermediate solutions in this continuous family include also soliton solutions for zero boundary conditions spectrally encoded by a single complex-conjugate pair of poles of arbitrary order, as well as other solutions having nonzero boundary conditions matching those of the rogue waves albeit with far slower decay as x → ± ∞ x\to \pm \infty . The large-order far-field asymptotic behavior of the solution depends on which of three disjoint regions C \mathcal {C} (the “channels”), S \mathcal {S} (the “shelves”), and E \mathcal {E} (the “exterior domain”) contains the rescaled variables. On the region C \mathcal {C} , the amplitude is small and the solution is highly oscillatory, while on the region S \mathcal {S} , the solution is approximated by a modulated plane wave with a highly oscillatory correction term. The asymptotic behavior on these two domains is the same for all continuous orders. Assuming that the order belongs to the discrete sequence characteristic of rogue-wave solutions, the asymptotic behavior of the solution on the region E \mathcal {E} resembles that on S \mathcal {S} but without the oscillatory correction term. Solutions of other continuous orders behave quite differently on E \mathcal {E} .