Nth-order Rogue Wave Solutions Research Articles

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.

General breather and rogue wave solutions to the complex short pulse equation

In the present paper, we attempt to construct both the breather and rogue wave solutions to the focusing complex short pulse (fCSP) equation via the KP–Toda reduction method. Following a procedure of reducing the bilinear equations satisfied by tau functions of Kadomtsev–Petviashvili (KP)–Toda hierarchy to the ones of the fCSP equation with nonzero boundary condition, we first deduce the general breather solution of the fCSP equation starting from a specially arranged tau-function of the KP–Toda hierarchy, then we construct and prove the Nth order rogue wave solutions of the fCSP equation and express them in two different but equivalent forms of determinants. The dynamical behaviors of both the breather and rogue wave solutions are illustrated and analyzed.

Rogue waves, semirational rogue waves and W-shaped solitons in the three-level coupled Maxwell–Bloch equations

In this paper, the coupled Maxwell–Bloch equations which describe the propagation of two optical pulses in an optical medium with coherent three-level atoms are studied by Darboux transformation. Both the general nth-order rogue wave solution involving two different choices of multiple roots for the spectral characteristic equation and the multiparametric nth-order semirational solution are obtained in terms of Schur polynomials. The explicit rogue wave solutions and semirational solutions from first to second order are provided. In contrast to the bright, dark and four-petaled structures, some unusual patterns such as triple-hole, twisted-pair, composite four-petaled and composite dark rogue waves are put forward. Moreover, the interactions between dark–bright solitons and dark rogue waves and interactions between breathers and dark rogue waves are shown. Further, the higher-order nonlinear superposition modes which feature triple and quadruple temporal–spatial distributions are presented. Finally, the state transitions between rogue waves and W-shaped solitons are found where the modulation instability growth rate tends to zero under the low perturbation frequency. Particularly, the dark and double-peak W-shaped solitons are examined.

Formula omitted]th-order rogue wave solutions for a variable coefficient Schrödinger equation in inhomogeneous optical fibers

In investigation is a nonlinear Schrödinger equation with variable coefficients, which can describe pulse propagation in inhomogeneous optical fiber systems. The variable coefficients can used to control the dispersion and nonlinearity factors. Through extending Darboux transformation iteration algorithm, analytical Nth-order rogue wave solutions are obtained for the equation. Based on the two free variable coefficient function involved in the solutions, novel rogue waves can be excited by choosing them as specific functions. Polynomial and periodic functions are taken as examples to how to excite rich rogue wave solutions. The results show that as the orders of the rogue wave solutions increase, the spatio-temporal patterns become more complex, while the amplitudes also increase rapidly. These properties will play critical roles in high-capacity information transmission and signal amplification in optical systems.

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.

Rogue waves and modulation instability in an extended Manakov system

We obtain the general nth-order rogue wave solutions of the vector cubic-quintic nonlinear Schrodinger equation with self-steepening, alias extended Manakov system, by means of a nonrecursive Darboux transformation scheme. We show that in such a two-component system, owing to the presence of the self-steepening effect, there would emerge an anomalous Peregrine soliton state on one wave component whose peak can grow three times higher than its background level, at the expense of a heavy falling-off on the other wave component. We also demonstrate other interesting rogue wave dynamics such as coexisting, doublet, quartet, and sextet rogue waves, which depend on the choice of structural parameters. In addition, the modulation instability responsible for the formation of rogue waves is discussed, revealing the broad-range existence of rogue waves, irrespective of what dispersion situations considered.

Rogue waves for a (2+1)-dimensional Gross–Pitaevskii equation with time-varying trapping potential in the Bose–Einstein condensate

The Bose–Einstein condensates (BECs) are seen during the studies in atomic optics, cavity opto-mechanics, cavity quantum electrodynamics, black-hole astrophysics, laser optics and atomtronics. Under investigation in this paper is a (2+1)-dimensional Gross–Pitaevskii equation with time-varying trapping potential which describes the dynamics of a (2+1)-dimensional BEC. Based on the Kadomtsev–Petviashvili hierarchy reduction, we construct the bilinear forms and the Nth order rogue-wave solutions in terms of the Gramian. With the help of the analytic and graphic analysis, we exhibit the first- and second-order rogue waves under the influence of the strength of the interatomic interaction, α(t), and of Ω(t)=ωR̃∕ωZ, where t is the scaled time, ωR̃ and ωZ are the confinement frequencies in the radial and axial directions: When Ω(t)=0, the first-order rogue wave exhibits as an eye-shaped distribution; When Ω(t) is a periodic function, the rogue wave periodically raises; When Ω(t) is an exponential function, the rogue wave appears on the exponentially increasing background; When α(t) decreases, background and amplitude of the rogue wave both increase. The second-order rogue waves reach the maxima only once or three times when Ω(t) is a constant. With Ω(t) being the exponential function, the backgrounds of the second-order rogue waves exponentially increase with t increasing.

Lax pair, Darboux transformation and rogue waves for the three-coupled fourth-order nonlinear Schrödinger system in an alpha helical protein

ABSTRACT Complexes of proteins are central to certain cellular processes. Investigated in this paper is the three-coupled fourth-order nonlinear Schrödinger system, which is used for describing the alpha helical proteins with interspine coupling at the fourth order. With respect to the three-component amplitudes of the molecular excitation, we derive a Lax pair and construct the corresponding Nth-order Darboux transformation, where N is a positive integer. Three types of the Nth-order rogue wave solutions are obtained with the help of the matrix analysis method. The first-order rogue waves with each component containing one, two or three rogue waves are derived. We observe that the width of the first-order vector rational rogue wave along the distance axis increases with the value of the lattice parameter increasing, while the width of the first-order vector rational rogue wave along the time axis decreases with the value of the lattice parameter increasing. We present the second-order vector rational rogue waves with each component constituted by five, seven or nine rogue waves. Vector semi-rational rogue waves display the coexistence of the rogue waves and line/Y-shaped breathers.

Darboux transformations and rogue wave solutions of a generalized AB system for the geophysical flows

In this paper, we investigate a generalized AB system, which is used to describe certain baroclinic instability processes in the geophysical flows. For the two short waves and mean flow, we derive out the Darboux and generalized Darboux transformations, both relevant to the coefficient of the nonlinear term and coefficient related to the shear. When the coefficient of the nonlinear term is positive, with the generalized Darboux transformation, we present the algorithm to derive the Nth-order (N=1,2,…) rogue wave solutions. The first- and second-order rogue wave solutions are shown, where our first-order rogue waves are different from those in the existing literatures. The two short waves and mean flow are related to the coefficient of the nonlinear term under certain conditions; the coefficient related to the shear has a linear effect on the mean flow while has no effect on the two short waves. The Nth-order rogue wave solutions turn to be singular when the coefficient of the nonlinear term is negative.

Soliton, Breather and Rogue Wave Solutions for the Nonlinear Schrödinger Equation Coupled to a Multiple Self-Induced Transparency System**Supported by the National Natural Science Foundation of China under Grant Nos 11705290 and 11305060, and the China Postdoctoral Science Foundation under Grant No 2016M602252.

We derive an N-fold Darboux transformation for the nonlinear Schrödinger equation coupled to a multiple self-induced transparency system, which is applicable to optical fiber communications in the erbium-doped medium. The N-soliton, N-breather and Nth-order rogue wave solutions in the compact determinant representations are derived using the Darboux transformation and limit technique. Dynamics of such solutions from the first- to second-order ones are shown.

W-shaped soliton complexes and rogue-wave pattern transitions for the AB system

With the help of a general Nth-order rogue wave solution in a compact determinant form for the AB system, we illuminate that, by suitably choosing the wavenumber and frequency of the background wave of the component A, W-shaped soliton complex containing a fixed number of algebraic solitons merging or separating with each other exists in the component A, and rogue-wave pattern transition between the four-petaled structure and dark structure occurs in the component B. The more complicated rogue-wave pattern transitions due to the nonlinear superposition corresponding to the higher-order four-petaled rogue waves and higher-order dark rogue waves of fundamental, triangular and circular dynamical structures up to third order are demonstrated, respectively. The spectrum properties which are related to the rogue-wave pattern transitions are revealed.

Breathers and Rogue Waves for the Fourth-Order Nonlinear Schrödinger Equation

Abstract: In this article, a generalized Darboux transformation for the fourth-order nonlinear Schrödinger equation is constructed in terms of Darboux matrix method. Subsequently, breathers and the Nth-order rogue wave solutions of this equation are explicitly given in the light of the obtained Darboux transformation. Finally, we concretely discuss the dynamics of the obtained rogue waves, which are also demonstrated by some figures.

Darboux transformation and rogue wave solutions for the variable-coefficients coupled Hirota equations

We construct the Lax pair and Darboux transformation for the variable-coefficients coupled Hirota equations. Based on modulation instability and by taking the limit approach, we derive two types of Nth-order rogue wave solutions with different dynamic structures in compact determinant representations. The explicit first-order rogue wave solution is presented, prolific vector rogue-wave patterns such as the dark-bright, composite, three-sister, quadruple and sextuple rogue waves with multiple compression points are demonstrated. In particular, in contrast to the standard Peregrine combs, unusual vector rogue wave combs such as the dark-bright and composite rogue wave combs are revealed by choosing sufficiently large periodic modulation amplitudes. Further, some wave characteristics such as the difference between light intensity and continuous wave background, and pulse energy evolution of the dark rogue wave solution that features multiple compression points are discussed in detail.

Soliton and rogue-wave solutions for a (2 + 1)-dimensional fourth-order nonlinear Schrödinger equation in a Heisenberg ferromagnetic spin chain

In this paper, we investigate the soliton and rogue-wave solutions for a (2 + 1)-dimensional fourth-order nonlinear Schrodinger equation, which describes the spin dynamics of a Heisenberg ferromagnetic spin chain with the bilinear and biquadratic interactions. For such an equation, there exists a gauge transformation which converts the nonzero potential Lax pair into some constant-coefficient differential equations. Solving those equations, vector solutions for the nonzero potential Lax pair are obtained. The condition for the modulation instability of the plane-wave solution is also given through the linear stability analysis. Then, we present the determinant representations for the N-soliton solutions via the Darboux transformation (DT) and Nth-order rogue-wave solutions via the generalized DT. Profiles for the solitons and rogue waves are analyzed with respect to the lattice parameter \(\sigma \), respectively. When \(\sigma \) is greater than a certain value marked as \(\sigma _{0}\), one-soliton velocities increase with the increase of \(\sigma \). When \(\sigma <\sigma _{0}\), one-soliton velocities decrease with the increase of \(\sigma \). When the time t is equal to zero, \(\sigma \) has no effect on the interactions between the two solitons. When \(t\ne 0\), different choices of \(\sigma \) lead to the different two-soliton velocities, giving rise to the different interaction regions. Widths of the first-order rogue waves become bigger with the decrease of \(\sigma \), while the amplitudes do not depend on \(\sigma \). The second-order rogue waves are composed by three first-order rogue waves whose widths all get wider with the decrease of \(\sigma \), while the amplitudes do not depend on \(\sigma \).

Rogue waves for a coupled nonlinear Schrödinger system in a multi-mode fibre

In this paper, we investigate the rogue waves for an integrable coupled nonlinear Schrödinger (CNLS) system with the self-phase modulation, cross-phase modulation and four-wave mixing term, which can describe the propagation of optical waves in a multi-mode fibre. We construct a generalized Darboux transformation (GDT) for the CNLS system and find a gauge transformation which converts the Lax pair into the constant-coefficient differential equations. Solving those equations, we can obtain the vector solutions of the Lax pair. Using the GDT, we derive an iterative formula for the nth-order rogue-wave solutions for the CNLS system. We derive the first- and second-order rogue-wave solutions for the CNLS system and analyse the profiles for the rogue waves with respect to the self-phase modulation term a, cross-phase modulation term c and four-wave mixing term b, respectively. The rogue waves become thinner with the increase in the value for the real part of b and that the effect of a or c on the rogue waves is the same as the one of the real part of b.

Higher-order rogue wave solutions of the three-wave resonant interaction equation via the generalized Darboux transformation

In this paper, we utilize generalized Darboux transformation to study higher-order rogue wave solutions of the three-wave resonant interaction equation, which describes the propagation and mixing of waves with different frequencies in weakly nonlinear dispersive media. A general Nth-order rogue wave solution with two characteristic velocities structural parameters and 3N independent parameters under a determined plane-wave background and a specific parameter condition is derived. As an application, we show that four fundamental rogue waves with fundamental, two kinds of line and quadrilateral patterns, or six fundamental rogue waves with fundamental, triangular, two kinds of quadrilateral and circular patterns can emerge in the second-order rogue waves. Moreover, several important wave characteristics including the maximum values, the corresponding coordinate positions of the humps, and the stability problem for some special higher-order rogue wave solutions such as the fundamental and quadrilateral cases are discussed.

Higher-order rogue wave solutions of the Kundu–Eckhaus equation

In this paper, we investigate higher-order rogue wave solutions of the Kundu–Eckhaus equation, which contains quintic nonlinearity and the Raman effect in nonlinear optics. By means of a gauge transformation, the Kundu–Eckhaus equation is converted to an extended nonlinear Schrödinger equation. We derive the Lax pair, the generalized Darboux transformation, and the Nth-order rogue wave solution for the extended nonlinear Schrödinger equation. Then, by using the gauge transformation between the two equations, a concise unified formula of the Nth-order rogue wave solution with several free parameters for the Kundu–Eckhaus equation is obtained. In particular, based on symbolic computation, explicit rogue wave solutions to the Kundu–Eckhaus equation from the first to the third order are presented. Some figures illustrate dynamic structures of the rogue waves from the first to the fourth order. Moreover, through numerical calculations and plots, we show that the quintic and Raman-effect nonlinear terms affect the spatial distributions of the humps in higher-order rogue waves, although the amplitudes and the time of appearance of the humps are unchanged.

Rogue wave solutions of AB system

In this paper, the generalized Darboux transformation is established to the AB system, which mainly describes marginally unstable baroclinic wave packets in geophysical fluids and ultra-short pulses in nonlinear optics. A unified formula of Nth-order rogue wave solution for the AB system is found by the direct iterative rule. In particular, rogue wave solutions possessing several free parameters from first to second order are calculated. The dynamic properties of rogue waves in the AB system are shown through some figures.

Generalized Darboux transformation and Nth order rogue wave solution of a general coupled nonlinear Schrödinger equations

We construct a generalized Darboux transformation (GDT) of a general coupled nonlinear Schrödinger (GCNLS) system. Using GDT method we derive a recursive formula and present determinant representations for Nth order rogue wave solution of this system. Using these representations we derive first, second and third order rogue wave solutions with certain free parameters. By varying these free parameters we demonstrate the formation of triplet, triangle and hexagonal patterns of rogue waves.

Generalized Darboux transformation and rogue wave solutions for the higher-order dispersive nonlinear Schrödinger equation

In this paper, rogue wave solutions of the higher-order dispersive nonlinear Schrödinger equation are investigated, which describe the propagation of ultrashort optical pulse in optical fibers. The Nth-order rogue wave solutions with 2N + 1 free complex parameters are constructed via the generalized Darboux transformation method. As applications, rogue waves from the first to the fifth order are calculated according to different combinations of parameters. In particular, rogue waves dynamics and several new spatial–temporal structures are also discussed and exhibited to make a comparison with those of the nonlinear Schrödinger equation.