Higher-order Rogue Waves Research Articles

Higher-order rogue wave solutions of the Kadomtsev Petviashvili—Benjanim Bona Mahony (KP-BBM) model via the Hirota-bilinear approach

We study the higher-order rogue wave solutions of the Kadomtsev Petviashvili—Benjanim Bona Mahony (KP-BBM) model with and without controllable center via the Hirota bilinear approach. To construct higher-order rogue wave solutions of the model, we apply cross-product terms in a polynomial expression as a test function. We construct three kinds of rogue waves with center at the origin and three kinds of higher-order rogue waves with a controllable center by choosing different test functions. We show how the center can control the shapes and orders of the rogue waves by choosing the values of parameters of the center. In particular, the 3-rogue wave solutions exhibit double rogue waves for lower values and distinct triple rogue waves in a triangular structure for a sufficiently large value parameters of the center. Moreover, the 6-rogue wave solutions of the model present lower-order rogue waves for small values and higher-order rogue waves for large value parameters of the center. It is shown that the order of the rogue gradually increases for rising value of parameters, and it will be maximum six distinct rogues for a sufficiently large values. Finally, we explain the solutions of the model in 3D and in density plots graphically.

Dynamics of higher-order bright and dark rogue waves in a new (2+1)-dimensional integrable Boussinesq model

This work deals with the dynamics of higher-order rogue waves in a new integrable (2+1)-dimensional Boussinesq equation governing the evolution of high and steep gravity water waves. To achieve this objective, we construct rogue wave solutions by employing Bell polynomial and Hirota’s bilinearization method, along with the generalized polynomial function. Through the obtained rogue wave solutions, we explore the impact of various system and solution parameters in their dynamics. Primarily, these parameters determine the characteristics of rogue waves, including the identification of their type, bright or dark type doubly-localized rogue wave structures and spatially localized rational solitons, and manipulation of their amplitude, depth, and width. Reported results will be encouraging to the studies on the rogue waves in higher dimensional systems as well as to experimental investigations on the controlling mechanism of rogue waves in optical systems, atomic condensates, and deep water oceanic waves.

Higher-order matter rogue waves and their deformations in two-component Bose–Einstein condensates

We construct matter rogue wave (RW) solutions for both attractive-attractive and repulsive-repulsive two-component Bose–Einstein condensates (BECs). With the help of similarity transformation, we transform the two coupled Gross–Pitaevskii (GP) equations into Manakov equation with an integrability condition. For attractive-attractive two-component BECs, we obtain various localized patterns, namely first-order dark-bright RW, doublets, triplets, quartets and sextet RWs. We explore the dynamical properties of the obtained RW solutions by considering three distinct forms of trap potentials, namely the time-independent trap, time-dependent monotonic trap and time-dependent periodic trap. Further, we derive first-order dark-bright, dark-dark doublet, dark-bright triplet and dark-dark triplet RW solutions for two-component BECs with repulsive interactions. We then analyze the deformation of these solutions for the same three kinds of trap potentials. Our results show that the robustness of the matter RWs can be manipulated with the help of the trap parameter.

Dynamical interactions between higher-order rogue waves and various forms of n-soliton solutions (n → ∞) of the (2+1)-dimensional ANNV equation

We present new lemmas, theorem and corollaries to construct interactions among higher-order rogue waves, n-periodic waves and n-solitons solutions (n → ∞) to the (2+1)-dimensional asymmetric Nizhnik–Novikov–Veselov (ANNV) equation. Several examples for theories are given by choosing definite interactions of the wave solutions for the model. In particular, we exhibit dynamical interactions between a rogue and a cross bright-dark bell wave, a rogue and a cross-bright bell wave, a rogue and a one-, two-, three-, four-periodic wave. In addition, we also present multi-types interactions between a rogue and a periodic cross-bright bell wave, a rogue and a periodic cross-bright-bark bell wave. Finally, we physically explain such interaction solutions of the model in the 3D and density plots.

Dynamics of localized wave solutions for the coupled Higgs field equation

By virtue of Hirota bilinear form of the coupled Higgs field equation, some higher-order rogue wave, the Ma breather, the Akhmediev breather and the general breather solutions are constructed through symbolic computation. With the help of the contour line method, we investigate the localization characters of the first-order rogue wave, which is different from the case of its in nonlinear Schrodinger equation. The Ma breather and the Akhmediev breather solutions of the coupled Higgs field equation possess time-periodic and space-periodic, respectively. The results of this paper show us the generation and evolutions of novel nontraveling wave with several interesting structures.

A matrix Yajima–Oikawa long-wave-short-wave resonance equation, Darboux transformations and rogue wave solutions

Based on an introduced (2m+n)×(2m+n) matrix spectral problem, a matrix Yajima–Oikawa long-wave-short-wave resonance equation is proposed, which can be reduced to an (n+1)-component Yajima–Oikawa long-wave-short-wave resonance equation. Multi-fold generalized Darboux transformations for these two equations are constructed by using the gauge transformation between Lax pairs and their Riccati equations. Every solution to the Riccati equations can be transformed into a new solution of the matrix Yajima–Oikawa long-wave-short-wave resonance equation through the Darboux transformations. As an application of the obtained Darboux transformation, first, we derive high-order rogue wave solutions of the Yajima–Oikawa long-wave-short-wave resonance equation. Second, we obtain explicit solutions for the lower-order matrix Yajima–Oikawa long-wave-short-wave resonance equation, including soliton solutions, rogue wave solutions, wave solutions that are constant along the x-direction, and wave solutions traveling at varying speeds.

Deformed breather and rogue waves for the inhomogeneous fourth-order nonlinear Schrödinger equation in alpha-helical proteins

The inhomogeneous fourth-order nonlinear Schrodinger equation is investigated, which models the transport of energy along the inhomogeneous hydrogen bonding spines in alpha-helical proteins. The deformed breather and rogue waves solutions for the equation are derived via Darboux transformation. Thereinto, the modified limit procedure in generalized Darboux transformation is proposed to construct the expressions of higher-order rogue waves. Due to the presence of inhomogeneity, despite their diversity profiles, the shape, amplitude and pulse width of each deformed breather will change with time. When the coefficients of higher-order terms are varied, the propagating trajectories of breathers are changed. Head-on collision, partially coalesced, without interaction between two deformed breathers is found under diverse selected parameters. The temporal-spatial structures of the presented rogue waves exhibit certain valleys around one or several centers, and the peak heights of those rogue waves are more than three times that of the background. Additionally, the distributed area of the rogue wave will tend to be broaden/narrow along the temporal axis, with the variation of the coefficients of inhomogeneity and higher-order terms.

Rogue wave solutions for a higher-order nonlinear Schrödinger equation in an optical fiber

Under investigation in this paper is a higher-order nonlinear Schrödinger equation in an optical fiber. Based on the symbolic computation and generalized Darboux transformation, we derive the rogue wave solutions for this equation. Propagation and interaction properties of the rogue waves are discussed. We see that peaks of the higher-order rogue waves could change to different shapes, and the rogue waves will be split under certain variable coefficients.

High-order rogue waves excited from multi-Gaussian perturbations on a continuous wave.

Peregrine rogue wave excitation has applications in gaining high-intensity pulses, etc., and a high-order rogue wave exhibits higher intensity. An exact solution and collision between breathers are two existing ways to excite high-order ones. Here we numerically report a new, to the best of our knowledge, possible method, which is by multi-Gaussian perturbations on a continuous wave. The order and maximal intensity of rogue waves can be adjusted by the number of perturbations. The maximal intensity approaches 63.8 times that of the power of the initial background wave, and it retains a large value under the influence of fiber loss and noise. Our results provide guidance in gaining high-intensity pulses in experiment and understanding the universality of rogue wave generation.

Fundamental Peregrine Solitons of Ultrastrong Amplitude Enhancement through Self-Steepening in Vector Nonlinear Systems.

We report the universal emergence of anomalous fundamental Peregrine solitons, which can exhibit an unprecedentedly ultrahigh peak amplitude comparable to any higher-order rogue wave events, in the vector derivative nonlinear Schrödinger system involving the self-steepening effect. We present the exact explicit rational solutions on either a continuous-wave or a periodical-wave background, for a broad range of parameters. We numerically confirm the buildup of anomalous Peregrine solitons from strong initial harmonic perturbations, despite the onset of competing modulation instability. Our results may stimulate the experimental study of such Peregrine soliton anomaly in birefringent crystals or other similar vector systems.

High-order rational solutions and rogue wave for the (2 + 1)-dimensional nonlinear Schrödinger equation

In this paper, some exact explicit rational solutions of the (2 + 1)-dimensional nonlinear Schrödinger equation are presented in terms of the Gram determinants by using the bilinear method. The expressions of the fundamental line rogue wave, second-order parallel line rogue wave and third-order parallel line rogue wave all involve determinants whose matrix elements are simple polynomials. These line rogue waves, which all generate from a constant background with line contours and then disappear into the same background, are plotted in the (x, y)-plane. Moreover, we also consider different structures of higher-order rogue waves, and their dynamical behaviors are illustrated in the (x, t)-plane.

Characteristics of higher-order vector rogue waves to a coupled fourth-order nonlinear Schrödinger system in a two-mode optical fiber

Optical fiber communication serves a significant role in recent communication. Investigated in the work is a coupled fourth-order nonlinear Schrodinger system for the ultrashort optical pulses in a two-mode optical fiber. For the complex envelopes of two field polarization components, we work out the Darboux transformation and higher-order vector rogue-wave solutions. The second-order vector rogue waves with the triangle structure, each component of which is composed of three four-petaled rogue waves, are shown. With the strength of the higher-order linear and nonlinear effects increasing, width of the triangle structure along the distance axis decreases, while width of the triangle structure along the time axis increases. We obtain the second-order vector rogue waves with the merged structure and the third-order vector rogue waves with the triangle and pentagon structures. We expect that our results could be helpful in studying the rogue-wave phenomena in the nonlinear optics.

High-order rogue waves of the generalized (3+1)-dimensional nonlinear wave in liquid with gas bubbles

High-order rogue waves of the generalized (3+1)-dimensional nonlinear wave in liquid with gas bubbles

Nonlinear dynamics of higher-order rogue waves in a novel complex nonlinear wave equation

A novel complex nonlinear wave equation was recently found by Mukherjee and Kundu (Phys. Lett. A 383:985–990, 2019) and shown that it possesses the first-order rogue waves and accelerated one-soliton solutions. In this paper, higher-order rogue wave solutions with multi-parameters of the novel complex nonlinear wave equation are derived by a symbolic computation approach. Nonlinear dynamics of the first- and second-order rogue wave solutions, localized in space–time and richer due to the presence of free parameters, are investigated in detail. In particular, a complete classification of the first-order rogue wave is given by the free parameters. With the help of the contour line method, some localization characters of the first-order rogue wave solution are analyzed. Moreover, the novel equation also allows some periodic wave and accelerated periodic wave solutions expressed by Jacobi elliptical functions.

Nonlinear Dynamics of Rogue Waves in a Fifth-Order Nonlinear Schrödinger Equation

In this paper, nonlinear dynamics of higher-order rogue waves are investigated for a fifth-order nonlinear Schrodinger equation, which can depict the Heisenberg ferromagnetic spin chain. A generalized Darboux transformation is constructed based on the Lax pair. Higher-order rogue waves solutions are given in terms of a recursive formula. Using numerical simulation, the first-order to the third-order rogue waves are displayed on the basis of some free parameters, which play a crucial role in affecting the distribution of rogue waves. The results obtained should be useful in understanding the generation mechanism of rogue waves.

Concurrent instabilities causing multiple rogue waves in infinite-dimensional dynamical systems

Complex instabilities are the major reason for drastic changes and extreme events in dynamical systems. Several modes of instability growing simultaneously with nonlinear interaction between them may lead to unforeseeable outcomes leading to catastrophic consequences. The most common examples of these instabilities are the modulation instability (MI). Studies show that an infinite number of instability modes remain active in a dynamical system. Although a one-mode MI can be analysed in the frame of a precise mathematical model, namely the Akhmediev breather, the dynamics of several concurrent MI modes referred to as the higher-order MI is very difficult to handle. We developed a unique geometrical approach that provides an entirely new and intuitive way to deal with higher-order MI. We apply this approach in description of higher-order modulation instability, multi-breather solutions, their degenerate versions and higher-order rogue waves of the nonlinear Schrodinger equation. For a system with infinitely many interacting instability modes, the band of the instability in this description is a hypercube, a multi-dimensional space of modulation frequencies. A large variety of special multi-breather and multi-rogue wave solutions of the nonlinear Schrodinger equation in this description corresponds to special points and lines within this hypercube.

High-order rogue waves of a long-wave-short-wave model of Newell type.

The long-wave-short-wave (LWSW) model of Newell type is an integrable model describing the interaction between the gravity wave (long wave) and the capillary wave (short wave) for the surface wave of deep water under certain resonance conditions. In the present paper, we are concerned with rogue-wave solutions to the LWSW model of Newell type. By combining the Hirota's bilinear method and the KP hierarchy reduction, we construct its general rational solution expressed by the determinant. It is found that the fundamental rogue wave for the short wave can be classified into three different patterns: bright, intermediate, and dark states, whereas the one for the long wave is always a bright state. The higher-order rogue wave corresponds to the superposition of fundamental ones. The modulation instability analysis shows that the condition of the baseband modulation instability where an unstable continuous-wave background corresponds to perturbations with infinitesimally small frequencies, coincides with the condition for the existence of rogue-wave solutions.

High-order rogue waves and their dynamics of the Fokas–Lenells equation revisited: a variable separation technique

The Fokas–Lenells (FL) equation is an integrable higher-order extension of nonlinear Schrodinger equation. One approach to generating its breather solutions is based on Darboux transformation (DT) and iterations. However, the DT of FL equation contains negative powers of the spectral parameter, which can lead to very complicated expressions when N is large. In this paper, we avoid the negative powers by adopting a variable separation and Taylor expansion technique to solve the Lax pair of FL system. Furthermore, stability of the proposed technique is demonstrated in detail.

Higher-order interactional solutions and rogue wave pairs for the coupled Lakshmanan–Porsezian–Daniel equations

The N-fold Darboux transformation of the coupled Lakshmanan–Porsezian–Daniel (LPD) equations is constructed. Based on the Darboux transformation and the limiting technique, we investigate two kinds of solutions for the coupled LPD equations, which are higher-order interactional solutions and rogue wave (RW) pairs. Through considering the double-root situation of the spectral characteristic equation for the matrix in the Lax pair, we give the higher-order interactional solutions among higher-order RWs, multi-bright (dark) solitons and multi-breather. Besides, we consider the triple-root situation of the spectral characteristic equation and get the higher-order RW pairs. It demonstrates that the RW pairs are greatly different form the traditional higher-order RWs. The fist-order RW pairs can split into two traditional first-order RWs, and four or six traditional fundamental RWs can emerge from the second-order case. The corresponding dynamics of these explicit solutions are discussed in detail.

Formula omitted]-symmetric nonlocal Davey–Stewartson I equation: Soliton solutions with nonzero background

Various solutions to the PT-symmetric nonlocal Davey–Stewartson (DS) I equation with nonzero boundary condition are derived by constraining different tau functions of the Kadomtsev–Petviashvili hierarchy combined with the Hirota bilinear method. From the first type of tau functions of the nonlocal DS I equation, we construct: (a) general line soliton solutions sitting on either a constant background or on a background of periodic line waves and (b) general lump-type soliton solutions. We find two generic types of line solitons that we call usual line solitons and new-found ones. The usual line solitons exhibit elastic collisions, whereas the new-found ones, in the evolution process, change their waveforms from an antidark (dark) shape to a dark (antidark) one. The general lump-type soliton solutions describe the interaction between 2N-line solitons and 2N-lumps, which give rise to different dynamical scenarios: (i) fusion of line solitons and lumps into line solitons, (ii) fission of line solitons into lumps and line solitons, and (iii) a combination of fusion and fission processes. By constraining another type of tau functions combined with the long wave limit method, periodic line waves, rogue waves, and semi-rational solutions to the nonlocal DS I equation are obtained in terms of determinants whose matrix elements have simple algebraic expressions. Finally, different types of general solutions of the nonlocal nonlinear Schrödinger equation, namely general higher-order breathers and mixed solutions consisting of higher-order breathers and rogue waves sitting on either a constant background or on a background of periodic line waves are obtained as reductions of the corresponding solutions of the nonlocal DS I equation.