Higher-order Rogue Waves Research Articles

Generalized perturbation (n, N − n) fold Darboux transformation for a nonlocal Hirota equation with variable coefficients

In this paper, a nonlocal Hirota equation with variable coefficients is investigated by applying the generalized perturbation (n, N − n) fold Darboux transformation method and Taylor expansion method. Multi-soliton solutions are obtained when the seed solution is trivial, and multi-soliton solutions, multi-breather solutions, high-order rogue wave solutions and their interaction solutions are obtained when the seed solution is a plane wave solution. Especially, we get the interaction solution of soliton, breather and rogue wave solution. In addition, by choosing appropriate parameters, the dynamic behaviors of the obtained solution are explored.

Dynamical analysis of higher-order rogue waves on the various backgrounds for the reverse space–time Fokas–Lenells equation

In this paper, we will focus on the reverse space–time Fokas–Lenells equation. According to the Lax pair, the semi-degenerate Darboux transformation will be constructed by Taylor expansions. Based on the determinant form, some rogue wave solutions will be derived on the three types of backgrounds, including M-shaped periodic waves, mixed M-shaped periodic waves and breather, mixed single-periodic and double-periodic waves. And the rogue waves in various backgrounds can be transformed into the corresponding backgrounds waves with parameters selection. The dynamic features of those solutions will be discussed via graphical illustration. It is worth mentioning that these results are novel.

Rogue wave patterns of the Fokas-Lenells equation

In this work, we study the high-order rogue wave solution for the Fokas-Lenells equation using the Kadomtsev-Petviashvili (KP) reduction method. These rogue wave patterns consist of triangle, pentagon, heptagon, nonagon, which are analytically described by the root structures of the Yablonskii-Vorob'ev polynomial hierarchy. On the other hand, we also report the other types of rogue wave patterns including heart-shaped, fan-shaped, two-arc+triangle, arc+pentagon, etc., which are analytically described by the root structures of Adler-Moser polynomials. These polynomials are the generalizations of the Yablonskii-Vorob'ev polynomial hierarchy, because of the arbitrariness of complex parameter . In addition, these rogue wave patterns are formed by the Peregrine solitons undergoing dilation, rotation, stretch, shear and translation. We also compare the prediction solutions with the corresponding true solutions and show the good consistency between them.

Modulation instability and localized wave excitations for a higher-order modified self-steepening nonlinear Schrödinger equation in nonlinear optics

This paper investigates a higher-order modified nonlinear Schrödinger equation with higher-order dispersion and self-steepening effects, which can be used to study the dynamics of asymmetric and steepened optical pulse transmission in optical fibres. The modulation instability of the plane-wave solution has been analysed, and the state transition of the rogue waves under the higher-order dispersion effect is presented. The findings demonstrate that the self-steepening effect may also produce an asymmetric unstable frequency band. Combined with the modulation instability, the rogue wave, rational soliton and mixed interaction of localized waves have a quantitative relationship with plane-wave parameters. Various exact solutions can be accurately located and obtained according to the generalized Darboux transformation. The asymptotic analysis of rational solutions demonstrates the state transition of higher-order rogue waves. This paper illustrates the significance of modulation instability for studying the integrable systems by the Darboux transformation. It is a new guidance to solve the difficulty of exact excitation of asymmetric localized waves in complex integrable systems. The results have prospective applications and references for the emergence, amplification and compression of asymmetric pulses in optical experiments.

Annular rogue waves in whispering gallery mode optical resonators

Based on the variable-coefficient Lugiato-Lefever equation, nonlinear dynamics of rogue waves in whispering gallery mode optical resonators are studied. By means of the analytical solution in the absence of pump, dynamical behaviors of the rogue waves in the periodic system are discussed. Firstly, the first-order rogue wave under the condition of no pumping has obvious periodic nature, and the periodicity of width and amplitude of rogue wave can be consistent by controlling the wavefront curvature, and the energy of rogue wave remains stable under the small gain parameter, and the inverse diffraction effect makes the center position of rogue wave produce a small offset. In addition, the amplitudes of second-order rogue waves have more complicated periodicity and longer period lengths both with the good stability. At last, the dynamical behaviors of rogue wave after adding external excitation are investigated. Under the periodical external excitation with the lower intensity, rogue wave can achieve a longer stable transmission time, while the smaller wavefront curvature can effectively prolong the stable transmission time of rogue wave. Moreover, we investigate the influence of different forms of external excitation on rogue wave, and the periodical external excitation makes rogue waves more stable than the constant external excitation. These results extend the application of higher-order rogue waves in (2 + 1)-dimensional nonlinear systems, and have implications for the development of high-energy optical pulses.

Multiple nonlinear wave solutions of a generalized Heisenberg ferromagnet model and their interactions

Under investigation in this paper is a generalized Heisenberg ferromagnet (HF) equation which is named the Zhanbota-IIA equation. It is one of the integrable generalizations of the HF equation that plays an important role in nonlinear magnetization dynamics. Through the establishment of the N-fold Darboux transformation, a series of solutions will be obtained, including multi-solitons, one- and two-breathers, first- and higher-order rogue waves. Dynamic behaviors of those solutions will be analyzed, including several structures of rogue waves such as fundamental structure, triangular structure, ring structure and ring-fundamental structure, the coexistence of rogue waves and breathers, i.e. semi-rational solution and the interaction of two breathers.

Dynamics of general higher-order rogue waves in the two-component long wave–short wave model of Newell type

An integrable two component long-wave–short-wave (2-LS) model of Newell type governing the resonant interaction between two capillary waves with a common gravity wave is considered. The general higher order rogue wave (RW) solutions of this system are obtained by employing the bilinear KP hierarchy reduction method. Our study explores three different families of RWs depending upon the root structure of an algebraic equation characterizing them. Those are the Nth-order bounded rogue waves, mixed bounded (N1,N2)th-order rogue waves and degradable bounded (N¯1,N¯2)th-order rogue waves, respectively, where N,N1,N2 are arbitrary positive integer and N¯1,N¯2 are arbitrary non-negative integers. It is interesting to note that the fundamental rogue wave in one short wave (SW) component can attain a peak amplitude as high as more than six times the background amplitude while it is three times the background amplitude in the other SW component. We identify that the Nth-order bounded rogue waves are composed of N(N+1)2 fundamental rogue waves, the mixed bounded (N1,N2)th-order rogue waves comprise N1th-order bounded rogue waves mixing with another type N2th-order bounded rogue waves existing in different state, and the degradable bounded (N¯1,N¯2)th-order rogue waves consist of N¯12+N¯22−N¯1(N¯2−1) fundamental rogue waves. Non-trivial super rouge waves are reported.

Higher-order rogue waves and dispersive solitons of a novel P-type (3+1)-D evolution equation in soliton theory and nonlinear waves

Higher-order rogue waves and dispersive solitons of a novel P-type (3+1)-D evolution equation in soliton theory and nonlinear waves

Rogue wave formation scenarios for the focusing nonlinear Schrödinger equation with parabolic-profile initial data on a compact support.

We study the (1+1) focusing nonlinear Schrödinger equation for an initial condition with compactly supported parabolic profile and phase depending quadratically on the spatial coordinate. In the absence of dispersion, using the natural class of self-similar solutions, we provide a criterion for blowup in finite time, generalizing a result by Talanov etal. In the presence of dispersion, we numerically show that the same criterion determines, even beyond the semiclassical regime, whether the solution relaxes or develops a high-order rogue wave, whose onset time is predicted by the corresponding dispersionless catastrophe time. The sign of the chirp appears to determine the prevailing scenario among two competing mechanisms for rogue wave formation. For negative values, the numerical simulations are suggestive of the dispersive regularization of a gradient catastrophe described by Bertola and Tovbis for a different class of smooth, bell-shaped initial data. As the chirp becomes positive, the rogue wave seems to result from the interaction of counterpropagating dispersive dam break flows, as in the box problem recently studied by El, Khamis, and Tovbis. As the chirp and amplitude of the initial profile are relatively easy to manipulate in optical devices and water tank wave generators, we expect our observation to be relevant for experiments in nonlinear optics and fluid dynamics.

Rogue waves on the periodic wave background in the Kadomtsev–Petviashvili I equation

Rogue waves arising on the background of two families of periodic standing waves in the Kadomtsev–Petviashvili I (KPI) equation are investigated. By the nonlinearization of spectral problem and Darboux transformation approach, the rogue wave solutions of the KPI equation on the Jacobian elliptic functions dn and cn background are derived. Darboux transformation is also used to introduce trigonometric function periodic background for the higher-order rogue wave in the KPI equation. On the plane wave background, some rogue waves, breathers and hybrid waves in the KPI equation are obtained as a byproduct in this paper. Moreover, the results represented in this paper enrich the dynamics of (2+1) dimensional nonlinear wave equations.

Mix-training physics-informed neural networks for high-order rogue waves of cmKdV equation

Mix-training physics-informed neural networks for high-order rogue waves of cmKdV equation

Higher-order regulatable rogue wave and hybrid interaction patterns for a new discrete complex coupled mKdV equation associated with the fourth-order linear spectral problem

Higher-order regulatable rogue wave and hybrid interaction patterns for a new discrete complex coupled mKdV equation associated with the fourth-order linear spectral problem

Rogue wave solutions of (3+1)-dimensional Kadomtsev-Petviashvili equation by a direct limit method

On the bases of N-soliton solutions of Hirota’s bilinear method, high-order rogue wave solutions can be derived by a direct limit method. In this paper, a (3+1)-dimensional Kadomtsev-Petviashvili equation is taken to illustrate the process of obtaining rogue waves, that is, based on the long-wave limit method, rogue wave solutions are generated by reconstructing the phase parameters of N-solitons. Besides the fundamental pattern of rogue waves, the triangle or pentagon patterns are also obtained. Moreover, the different patterns of these solutions are determined by newly introduced parameters. In the end, the general form of N-order rogue wave solutions are proposed.

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.

Dynamic behaviors of vector breather waves and higher-order rogue waves in the coupled Gerdjikov–Ivanov equation

We extend Darboux-dressing transformation with expansion theorem to study the coupled Gerdjikov–Ivanov (cGI) equation. We successfully find its novel vector breather wave solutions and Nth-order rouge wave solutions. By considering an expansion theory, we first construct the Darboux-dressing transformation, which could iterate with the same spectral parameters for finding interesting exact solutions of the cGI equation. Based on the resulting Darboux-dressing transformation, we derive its exact breather wave solutions by means of matrix exponential function. Moreover, the Taylor multi-series expansion method is employed to derive higher-order rouge wave solutions of the equation. Finally, some interesting dynamic behaviors of breather waves and rouge waves are analyzed graphically.

Breathers and higher order rogue waves on the double-periodic background for the nonlocal Gerdjikov–Ivanov equation

Breathers and higher order rogue waves on the double-periodic background for the nonlocal Gerdjikov–Ivanov equation

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.

Solutions to the complex shifted reverse space-time modified Korteweg-de Vries equation

According to the new nonlocal integrable reduction proposed by Ablowitz and Musslimani, we study the complex shifted reverse space-time modified Korteweg-de Vries equation. The soliton, the kink-type breather and higher-order rogue waves solutions are constructed by using the Darboux transformation method. The effects of the spectral parameter and space-time shift parameters on the solution are discussed respectively, and the dynamic behavior of them is described from the expression and image of the solution. In addition, by comparing with the standard PT symmetry, we discuss the new structure of different solutions due to the space-time shift parameters.

General high-order rogue waves in the Hirota equation

In this work, we focus on the construction of Nth-order rogue waves for the Hirota equation by utilizing the bilinear method and Kadomtsev–Petviashvili (KP) hierarchy reduction technique. The formula can be represented in terms of determinants whose matrix elements are related to Schur polynomials. In addition, some interesting dynamic patterns of rogue waves are exhibited analytically and graphically.