Higher-order Rogue Wave Solutions Research Articles

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.

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.

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.

Multi-breather and high-order rogue waves for the quintic nonlinear Schrödinger equation on the elliptic function background

In this paper, we derive the multi-breather and higher-order rogue wave solutions of the quintic nonlinear Schrödinger (NLS) equation on the elliptic function background by using the Darboux transformation (DT) and the modified squared wave (MSW) function approach. Firstly, we construct the elliptic function solution and solve the corresponding associated Lax pair with the given elliptic function potential by the traveling wave transformation. Secondly, we use the theta functions to represent the elliptic function solution and the fundamental solution of the Lax pair. Thirdly, we present several types of breathers on the dn- and cn-periodic backgrounds including the general breather (GB), the Kuznetsov–Ma breather (KMB) and the Akhmediev breather (AB). In addition, we illustrate the nonlinear dynamics of the breathers on the effect of elliptic function background. Finally, at branch points for breather solutions we obtain the higher-order rogue wave solutions by using the generalized DT. By several examples, we provide the first-, second- and second second-order rogue waves on the elliptic function background. It is expected that our results are helpful to understand the dynamics of breathers and rogue waves on the periodic background in various physical systems with higher-order effects.

Higher-order rogue-wave fission in the presence of self-steepening and Raman self-frequency shift

Using the generalized nonlinear Schr\"odinger equation, we investigate how the effect of self-steepening and Raman-induced self-frequency shift impact higher-order rogue-wave solutions. We observe that each effect breaks apart the higher-order rogue wave, reducing it to its constituent fundamental parts, in a similar manner to how a higher-order soliton undergoes fission. Applying a local inverse-scattering technique, we show that under the effect of self-frequency shift, the emergence of a rogue wave significantly influences the surrounding wave background, triggering solitons, breathers, and new rogue waves. We demonstrate that under the combined effect of third-order dispersion, self-steepening, and Raman-induced self-frequency shift, the disintegrated elements of higher-order rogue waves become fundamental solitons, creating an asymmetrical spectral profile that generates both red- and blue-shifted frequency components. We also show the intermediary processes of the fission steps prior to soliton transformation which can only be observed in the presence of weak perturbations. These observations reveal the mechanisms that create a large number of solitons in the process of modulation instability-induced supercontinuum generation from a continuous-wave background in optical fibers.

Darboux transformation and general soliton solutions for the reverse space–time nonlocal short pulse equation

We construct the Darboux transformation for the reverse space–time (RST) nonlocal short pulse equation by considering a nonlocal symmetry reduction of the Ablowitz–Kaup–Newell–Segur spectral problem and a covariant hodograph transformation. Some essential differences between the RST nonlocal models and the usual (local) ones are demonstrated by means of a series of explicit solutions and the solving process. The multi-bright soliton solution on vanishing background, as well as the multi-dark soliton, multi-breather and higher-order rogue wave solutions corresponding to nonvanishing background of the RST nonlocal defocusing short pulse equation are derived through the Darboux transformation. The classification and dynamics of these explicit solutions with loop-, cuspon- and smooth-type are presented. The asymptotic analysis is rigorously performed for the two-bright soliton, two-dark soliton and two-breather solutions. Some novel wave patterns such as the double-loop and mixed loop-cuspon rogue waves which are different from their regular counterparts are shown. Other types of singular solutions along certain space–time lines are also obtained.

Higher-order rogue wave solutions of the (2+1)-dimensional Fokas–Lenells equation

In this paper, we present the nonlinear dynamic behaviors of the higher-order rogue wave solutions to the (2+1)-dimensional Fokas–Lenells equation. The (1+1)-dimensional Fokas–Lenells equation is generalized to the (2+1)-dimensional Fokas–Lenells one via the Lax pair approach. Moreover, infinitely many conservation laws and the determinant of n-fold generalized Darboux transformation are derived for the (2+1)-dimensional Fokas–Lenells equation. The modulation instability is also analyzed based on the theory of standard linear-stability. In addition, we obtain the higher-order rogue wave solutions of the (2+1)-dimensional Fokas–Lenells equation by the generalized Darboux transformation from an exponential seed solution. Furthermore, we plot the graph of rogue wave solutions and discuss in detail the variation of the rogue wave solutions with different parameters, which show interesting structures including triangles and pentagons.

Higher-order rogue waves solutions of the modified Gerdjikov–Ivanov equation with dispersion via generalized Darboux transformation

Higher-order rogue wave solutions have vital applications in nonlinear physics. The new modified Gerdjikov–Ivanov equation with dispersion is derived for the first time in this paper. Based on the extend Lax pair, the localized waves of the new model are obtained via generalized Darboux transformation, the higher-order rogue solutions are presented graphically and their dynamic properties are analyzed. The results obtained have spurred the study of higher-order dispersion equations in nonlinear optics.

A complex short pulse system in optical-fiber communications: Rogue waves and phase transitions

In this article, we investigate a complex short pulse system (CSPS) in optical-fiber communications with fast varying packets. Via the Darboux transformation technique, new higher-order rogue wave solutions are constructed for the system. Through controlling the parameter, novel and phase transitions from single-peak to multi-ring-folded rogue waves are observed. The results are helpful to understand the formation mechanism of rogue wave compressions.

Higher-order rogue wave solutions of the Sasa–Satsuma equation

Up to the third-order rogue wave solutions of the Sasa–Satsuma (SS) equation are derived based on the Hirota’s bilinear method and Kadomtsev–Petviashvili hierarchy reduction method. They are expressed explicitly by rational functions with both the numerator and denominator being the determinants of even order. Four types of intrinsic structures are recognized according to the number of zero-amplitude points. The first- and second-order rogue wave solutions agree with the solutions obtained so far by the Darboux transformation. In spite of the very complicated solution form compared with the ones of many other integrable equations, the third-order rogue waves exhibit two configurations: either a triangle or a distorted pentagon. Both the types and configurations of the third-order rogue waves are determined by different choices of free parameters. As the nonlinear Schrödinger equation is a limiting case of the SS equation, it is shown that the degeneration of the first-order rogue wave of the SS equation converges to the Peregrine soliton.

Modulational instability and position controllable discrete rogue waves with interaction phenomena in the semi-discrete complex coupled dispersionless system

The position controllable discrete analytical higher-order rogue wave and periodic wave solutions in the semi-discrete complex coupled dispersionless system, as well as their mixed interaction phenomena, are theoretically provided in this study. Our first priority is to study the modulation instability theory study for this system and deduce the formation mechanism from its plane wave solutions. Then, we use the generalized (n,N−n)-fold Darboux transformation to create three types of position-controlled localized wave solutions, including rogue waves, periodic waves and their mixed interaction solutions. In particular, we find three different kinds of rogue wave structures in the same discrete system, including dark–bright structure, bright–bright structure, and the structure with two peaks and two depressions, which is a rare phenomenon in the field of nonlinear waves. All these innovative structures are discussed analytically and shown graphically. Besides, the principal structures and locations of these novel solutions can be controlled by a set of unique parameters so that we can theoretically manage them to appear anywhere on the plane wave by changing these parameters. These results may play a potential role to describe the interaction between the current-fed string and the external magnetic field and other physical phenomena.

Painlevé analysis and higher-order rogue waves of a generalized (3+1)-dimensional shallow water wave equation

Considering the importance of ever-increasing interest in exploring localized waves, we investigate a generalized (3+1)-dimensional Hirota-Satsuma-Ito equation describing the unidirectional propagation of shallow-water waves and perform Painlevé analysis to understand its integrability nature. We construct the explicit form of higher-order rogue wave solutions by adopting Hirota’s bilinearization and generalized polynomial functions. Further, we explore their dynamics in detail, depicting different pattern formation that reveal potential advantages with available arbitrary constants in their manipulation mechanism. Particularly, we demonstrate the existence of singly-localized line-rogue waves and doubly-localized rogue waves with multiple (single, triple, and sextuple) structures generating triangular and pentagon type geometrical patterns with controllable orientations that can be altered appropriately by tuning the parameters. The presented analysis will be an essential inclusion in the context of rogue waves in higher-dimensional systems.

Dynamics of general higher-order rogue waves in the two-component nonlinear Schrödinger equation coupled to the Boussinesq equation

We report three families of general higher-order rogue wave solutions to the two-component nonlinear Schrödinger equation coupled to the Boussinesq (2NLS–Boussinesq) equation via the bilinear Kadomtsev–Petviashvili hierarchy reduction method, which governs two short waves (SWs) interacting with a common long wave (LW). In the first solution family, the fundamental rogue waves in the SW components admit three different states: bright, dark, and four-petaled ones, whereas they are always bright state in the LW component. The higher-order rogue waves are superpositions of N(N+1)2 bounded fundamental rogue waves. The second solution family can be regarded as an extension of the first family, which is a superposition of the N1th-order and N2th-order bounded rogue waves in the first solution family. These N1th-order and N2th-order bounded rogue waves are in different states, thus the rogue waves of composite states can be found, including bright–dark rogue waves, bright-four-petaled rogue waves, and dark-four-petaled rogue waves. The third family contains the degenerated solutions of the first family, and consists of [N̂12+N̂22−N̂1(N̂2−1)] bounded fundamental rogue waves. Here N,N1,N2 are positive integers, while N̂1 and N̂2 are nonnegative integers. Some degenerated rogue waves are exhibited, such as the special cases consisting of two, four, or five bounded fundamental rogue waves.

Generalized Darboux transformation and higher-order rogue wave solutions to the Manakov system

In this paper, we propose a recursive Darboux transformation in a generalized form of a focusing vector Nonlinear Schrödinger Equation (NLSE) known as the Manakov System. We apply this generalized recursive Darboux transformation to the Lax-pairs of this system in view of generating the Nth-order vector generalization rogue wave solutions with a rule of iteration. We discuss from first- to three-order vector generalizations of rogue wave solutions while illustrating these features with some 3D, 2D graphical depictions. We illustrate a clear connection between higher-order rogue wave solutions and their free parameters for better understanding the physical phenomena described by the Manakov system

Coupled derivative nonlinear Schrödinger III equation: Darboux transformation and higher-order rogue waves in a two-mode nonlinear fiber

The coupled derivative nonlinear Schrödinger III (cDNLSIII) equation describes pulse propagations in a two-mode nonlinear fiber. In this paper, we construct the N-fold generalized Darboux transformation (DT) with the limit approach, from which the general higher-order rogue wave solutions of the cDNLSIII equation are obtained. These solutions, divided into three categories, namely, rogue waves and rogue waves on multi-soliton and multi-breather backgrounds, have applications in nonlinear fiber communication. Furthermore, we present some interesting interactions between rogue waves and solitons or breathers and analyze the features of each type.

Higher-order rogue wave solutions to the Kadomtsev–Petviashvili 1 equation

We construct a family of eigenfunction solutions of the Lax pair of the Kadomtsev–Petviashvili 1 (KP1) equation, which is expressed in terms of the Taylor coefficients of a fundamental exponential function associated with the Lax pair. Using the binary Darboux transformation, the higher-order rogue waves on a solitonic background of the KP1 equation are obtained by multiple soliton interactions. In fact, these solitons can evolve to significant and strongly localized transient waves during directional evolution, similarly to KP2 shallow-water interactions. We conjecture that the nth-order rogue wave solution evolves in the form of a triangular extreme wave pattern that consists of n(n+1)2 solitonic lumps in the intermediate time, while only n+1 parallel line solitons possessing equal height are present before and after the initiation of collision dynamics. Such fascinating higher-order rogue waves have not yet been reported in the context of this 2+1 dimensional integrable system. These exact solutions are particularly relevant for the fundamental understanding of extreme wave events in a variety of physical systems, such as plasma and solids.

Modulation instability, higher-order rogue waves and dynamics of the Gerdjikov–Ivanov equation

We investigate the modulation instability and higher-order rogue waves for the Gerdjikov–Ivanov equation. Based on the theory of the linear stability analysis, the modulation instability is the condition of the existence of the rogue waves. With the help of Darboux transformation and a variable separation technique, the formula of the higher-order rogue wave solutions is given explicitly. The kinetics of the first-, second-, and third-order rogue wave solutions are elucidated from the viewpoint of three-dimensional structures. More specifically, it is shown that this method is fairly powerful and handy to obtain the higher-order rogue wave solutions which appear in different phenomena in applied sciences and mathematical physics.

Multi-rogue wave solutions for a generalized integrable discrete nonlinear Schrödinger equation with higher-order excitations

In this paper, we construct the discrete higher-order rogue wave (RW) solutions for a generalized integrable discrete nonlinear Schrödinger (NLS) equation. First, based on the modified Lax pair, the discrete version of generalized Darboux transformation is constructed. Second, the dynamical behaviors of first-, second- and third-order RW solutions are investigated in corresponding to the unique spectral parameter, higher-order term coefficient, and free constants. The differences between the RW solution of the higher-order discrete NLS equation and that of the Ablowitz–Ladik (AL) equation are illustrated in figures. Moreover, we explore the numerical experiments, which demonstrates that strong-interaction RWs are stabler than the weak-interaction RWs. Finally, the modulation instability of continuous waves is studied.

Higher-order hybrid waves for the (2 + 1)-dimensional Boiti–Leon–Manna–Pempinelli equation for an irrotational incompressible fluid via the modified Pfaffian technique

Fluids, as a phase of matter including liquids, gases and plasmas, are seen to be common in nature, the study of which helps the design in the related industries. In this paper, we optimize the Pfaffian technique and investigated the Boiti–Leon–Manna–Pempinelli equation for an irrotational incompressible fluid. Higher-order hybrid solutions consisting of the L lumps, M breathers and N solitons are constructed with L, M and N being positive integers. Relative extrema of the breather and lump are presented, respectively. Breather is found to be localized along the curve $$a_1 x+b_1 \varphi (y)+\omega _1 t+\xi _1=0$$ and periodic along the curve $$\alpha _1 x+\beta _1 \varphi (y)+\gamma _1 t+\theta _1=0$$ . Under the lump existence condition, higher-order rogue wave solutions do not exist. Hybrid solutions composed of breathers, lumps and solitons are illustrated graphically. It can be found that when certain parameters are chosen, the breather, lump and soliton included in the hybrid solutions possess the same properties as those of the breather and lump solutions.

Higher-order rogue wave solutions for a generalized Hirota-Maxwell-Bloch system in the erbium-doped fibers

Under investigation in this paper is a generalized Hirota-Maxwell-Bloch System, which describes the propagation of optical solitons in the erbium-doped medium. With symbolic computation, the generalized N-fold Darboux transformation (GDT) will be constructed. As the application of the GDT, some types of higher-order rogue waves will be presented accurately by choosing different parameters and dynamic behaviors of those rogue waves will be analyzed.