Third-order Rogue Waves Research Articles

Multiple rogue wave solutions of the (3+1)-dimensional Kadomtsev–Petviashvili–Boussinesq equation

Based on a symbolic computation approach and the Hirota’s bilinear method, the multiple rogue wave solutions of the (3+1)-dimensional Kadomtsev–Petviashvili–Boussinesq equation, which can be regarded as a generalization of the generalized rational solutions of Boussinesq equation proposed by Clarkson and Dowie, are constructed. The first-order, second-order, third-order and fourth-order rogue waves are systematically discussed. Moreover, the maximal amplitude and the minimum amplitude of the first-order rogue wave solutions are given. By choosing some specific parameters of these rogue wave solutions, their dynamic behaviors are analyzed.

Multiple rogue wave solutions for a (3+1)-dimensional Hirota bilinear equation

In this paper, we considered the multiple rogue wave solutions of a (3+1)-dimensional Hirota bilinear equation by using a symbolic computation approach. Based on the bilinear form of this equation, the first-order rogue waves, the second-order rogue waves and the third-order rogue waves are presented. Moreover, some basic properties of multiple rogue waves and their collision structures are explained by drawing the three dimensional plot.

Rogue waves for an inhomogeneous discrete nonlinear Schrödinger equation in a lattice

Lattices are used in such fields as electricity, optics and magnetism. Under investigation in this paper is an inhomogeneous discrete nonlinear Schrödinger equation, which models the wave propagation in a lattice. Employing the Kadomtsev–Petviashvili (KP) hierarchy reduction, we obtain the rogue-wave solutions, and see that the rogue waves are affected by the coefficient of the on-site external potential. We see (1) the first-order rogue wave with one peak and two hollows; (2) the second-order rogue waves, each of which is with one peak or three humps; (3) the third-order rogue waves, each of which is with one peak or six humps, and those humps exhibit the triangular pattern, anti-triangular pattern and circular pattern. When the coefficient of the on-site external potential is a constant, the rogue waves periodically appear. When the coefficient of the on-site external potential monotonously changes, oscillations emerge on the constant background.

Dynamics of Rogue Waves for a Generalized Inhomogeneous Third-Order Nonlinear Schrödinger Equation From the Heisenberg Ferromagnetic System

In this paper, dynamics of the higher-order rogue waves for a generalized inhomogeneous third-order nonlinear Schrodinger equation is investigated by using the generalized Darboux transformation. Based on the Lax pair, the first-order to the third-order rogue wave solutions are derived through algebraic iteration starting from a seed solution. Nonlinear dynamical properties of rogue waves are analyzed on the basis of 3-D plots and density profiles. The new arrangement of the higher-order rogue waves is obtained. It is helpful to study the phenomenon of rogue waves in the Heisenberg ferromagnetic system.

Breathers and rogue waves in a ferromagnetic thin film with the Dzyaloshinskii-Moriya interaction

We report a theoretical research on nonlinear localized excitations in a ferromagnetic thin film with an interfacial Dzyaloshinskii-Moriya interaction. Employing the Generalized Darboux transformation, analytical forms for magnetic breathers and the first- to third-order rogue wave solutions are exactly constructed. It is found that the introduction of the Dzyaloshinskii-Moriya interaction may lead to a reverse effect of the propagation direction of the magnetic breather, and varying the value of the Dzyaloshinskii-Moriya interaction parameter can change the velocity of the magnetic breather. Furthermore, our results display that first- to third-order magnetic rogue waves will rotate clockwise as the intensity of the Dzyaloshinskii-Moriya interaction increases.

Lax pair, Darboux transformation and [formula omitted]th-order rogue wave solutions for a (2+1)-dimensional Heisenberg ferromagnetic spin chain equation

A new (2+1)-dimensional Heisenberg ferromagnetic spin chain equation is investigated, which can be used to describe magnetic soliton excitations in two dimensional space fields and a time field. The Lax pair of the equation is first constructed. Based on the Lax pair, initial seed solution and Darboux transformation, the analytic first-, second- and third-order rogue wave solutions are obtained, and a general expression of the Nth-order (N>3) rogue wave solutions is presented. The impacts of the system parameters on the rogue waves are demonstrated through numerical visualization method.

Rogue-wave solutions for a discrete Ablowitz–Ladik equation with variable coefficients for an electrical lattice

We investigate a discrete Ablowitz–Ladik equation with variable coefficients, which models the modulated waves in an electrical lattice. Employing the similarity transformation and Kadomtsev–Petviashvili hierarchy reduction, we obtain the rogue-wave solutions in the Gram determinant form under certain variable-coefficient constraints. We graphically study the rogue waves with the influence of the coefficient of tunnel coupling between the sites, $$|\varLambda (t)|$$ , time-modulated effective gain/loss term, $$\gamma (t)$$ , space–time-modulated inhomogeneous frequency shift, $$v_n(t)$$ ( $$n=1,2,\ldots $$ ), and lattice spacing, h, where t is the scaled time. Increasing value of h leads to the decrease in the rogue waves’ amplitudes. Properties of the rogue waves with $$|\varLambda (t)|$$ as the polynomial, sinusoidal, hyperbolic and exponential functions are discussed, respectively. The monotonically increasing, monotonically decreasing, periodic and Gaussian backgrounds are, respectively, displayed with the different $$\gamma (t)$$ . The first-order rogue wave exhibits one hump and two valleys, and the second-order rogue waves exhibit three humps and one highest peak. The third-order rogue waves with the six humps and one highest peak are also presented.

General high-order rogue waves to nonlinear Schrödinger–Boussinesq equation with the dynamical analysis

General high-order rogue waves of the nonlinear Schrodinger–Boussinesq equation are obtained by the KP-hierarchy reduction theory, and the N-order rogue waves are expressed with the determinants, whose entries are all algebraic forms, which is shown in the theorem. It is found that the fundamental first-order rogue waves can be classified into three patterns: four-petal state, dark state, bright state by choosing different values of parameter $$\alpha $$ . An interesting phenomenon is discovered as the evolution of the parameter $$\alpha $$ : the rogue wave changes from four-petal state to dark state, whereafter bright state, which are consistent with the change in the corresponding critical points to the function of two variables. Furthermore, the dynamical property of second-order and third-order rogue waves is plotted, which can be regarded as the nonlinear superposition of the fundamental first-order rogue waves.

Rogue waves for a discrete (2+1)-dimensional Ablowitz-Ladik equation in the nonlinear optics and Bose-Einstein condensation

Under investigation in this paper is a discrete (2+1)-dimensional Ablowitz-Ladik equation, which is used to model the nonlinear waves in the nonlinear optics and Bose-Einstein condensation. Employing the Kadomtsev-Petviashvili hierarchy reduction, we obtain the rogue wave solutions in terms of the Gramian. We graphically study the first-, second- and third-order rogue waves with the influence of the focusing coefficient and coupling strength. When the value of the focusing coefficient increases, both the peak of the rogue wave and background decrease. When the value of the coupling strength increases, the rogue wave raises and decays in a shorter time. High-order rogue waves are exhibited as one single highest peak and some lower humps, and such lower humps are shown as the triangular and circular patterns.

Conservation laws, modulation instability and rogue waves for the localized magnetization with spin torque

Under investigation in this paper is an integrable equation which can describe the localized magnetization with spin torque under the long-wavelength approximation. In order to obtain the rogue wave solutions, a new Lax pair is derived. Infinitely-many conservation laws are also constructed. Based on the generalized Darboux transformation, the first-, second- and third-order rogue wave solutions are derived. Property of the rogue waves are analyzed and influence of parameters α, β and separating function f(ϵ) on the rogue waves and spatial-temporal structures are also discussed (the meaning of α and β can be found in the paper). For the case of f(ϵ)=0, the modulus of the kth-order rogue wave (k=1,2,3) is irrelevant to parameter α. Parameter β influences the spatial-temporal range where the rogue wave appears. Spatial-temporal range enlarges with the increase of β. In addition, β also produces a skew angle and the skew angle rotates in the counter clockwise direction with the increase of β. f(ϵ) influences the spatial-temporal structures of the second- and third-order rouge waves. If f(ϵ) ≠ 0, the second-order rogue wave will split into three single first-order rouge waves and the triangular pattern can be formed, while the third-order rogue wave will split into six ones and the triangular pattern and pentagon pattern can be formed. The linear stability analysis is carried out, which shows that the modulation instability process is influenced by the amplitude of the harmonic wave and the wave number.

Rogue Wave Solutions and Generalized Darboux Transformation for an Inhomogeneous Fifth-Order Nonlinear Schrödinger Equation

The rogue wave solutions are discussed for an inhomogeneous fifth-order nonlinear Schrödinger equation, which describes the dynamics of a site-dependent Heisenberg ferromagnetic spin chain. Using the Darboux matrix, the generalized Darboux transformation is constructed and a recursive formula is derived. Based on the transformation, the first-order to the third-order rogue wave solutions are obtained. Then, the nonlinear dynamics of the first-order to the third-order rogue waves are studied on the basis of some free parameters. Several new structures of the rogue waves are found using numerical simulation. The conclusions will be a supportive tool to study the rogue waves better.

Soliton, Breather, and Rogue Wave for a (2+1)-Dimensional Nonlinear Schrödinger Equation

Abstract In this paper, a (2+1)-dimensional nonlinear Schrödinger (NLS) equation, which is a generalisation of the NLS equation, is under investigation. The classical and generalised N-fold Darboux transformations are constructed in terms of determinant representations. With the non-vanishing background and iterated formula, a family of the analytical solutions of the (2+1)-dimensional NLS equation are systematically generated, including the bright-line solitons, breathers, and rogue waves. The interaction mechanisms between two bright-line solitons and among three bright-line solitons are both elastic. Several patterns for first-, second, and higher-order rogue wave solutions fixed at space are displayed, namely, the fundamental pattern, triangular pattern, and circular pattern. The two-dimensional space structures of first-, second-, and third-order rogue waves fixed at time are also demonstrated.

Rogue-Wave Interaction of a Nonlinear Schrödinger Model for the Alpha Helical Protein

Abstract In this article, we investigate a fourth-order nonlinear Schrödinger equation, which governs the Davydov solitons in the alpha helical protein with higher-order effects. By virtue of the generalised Darboux transformation, higher-order rogue-wave solutions are derived. Propagation and interaction of the rogue waves are analysed: (i) Coefficients affect the existence time of the first-order rogue waves; (ii) coefficients affect the interaction time of the second- and third-order rogue waves; (iii) direction of the rogue-wave propagation remain unchanged after interaction.

Solitons and Rogue Waves for a Higher-Order Nonlinear Schrödinger–Maxwell–Bloch System in an Erbium-Doped Fiber

AbstractUnder investigation in this article is a higher-order nonlinear Schrödinger–Maxwell–Bloch (HNLS-MB) system for the optical pulse propagation in an erbium-doped fiber. Lax pair, Darboux transformation (DT), and generalised DT for the HNLS-MB system are constructed. Soliton solutions and rogue wave solutions are derived based on the DT and generalised DT, respectively. Properties of the solitons and rogue waves are graphically presented. The third-order dispersion parameter, fourth-order dispersion parameter, and frequency detuning all influence the characteristic lines and velocities of the solitons. The frequency detuning also affects the amplitudes of solitons. The separating function has no effect on the properties of the first-order rogue waves, except for the locations where the first-order rogue waves appear. The third-order dispersion parameter affects the propagation directions and shapes of the rogue waves. The frequency detuning influences the rogue-wave types of the module for the measure of polarization of resonant medium and the extant population inversion. The fourth-order dispersion parameter impacts the rogue-wave interaction range and also has an effect on the rogue-wave type of the extant population inversion. The value of separating function affects the spatial-temporal separation of constituting elementary rogue waves for the second-order and third-order rogue waves. The second-order and third-order rogue waves can exhibit the triangular and pentagon patterns under different choices of separating functions.

Breather and rogue wave solutions for a nonlinear Schrödinger-type system in plasmas

A nonlinear Schrodinger equation with self-consistent sources can be used to describe the interaction between an high-frequency electrostatic wave and an ion-acoustic wave in two-component homogeneous plasmas. In this paper, breather and rogue waves solutions for such a system are investigated via the Darboux transformation. The spatially and temporally periodic breathers have been found. When the cubic nonlinearity coefficient and the background amplitude are varied, the propagating trajectories of breather are changed. The interaction between two breathers are also studied, which shows that when the wave numbers are close, the two breathers are partially coalesced. All of the first-, second-, and third-order rogue waves present certain lumps or valleys around a center in the temporal–spatial distribution, and the peak heights of the ion-acoustic waves are more than three times those of the background. However, for the high-frequency electrostatic waves, the values around the centers are less than the background intensity.

Manipulating matter rogue waves and breathers in Bose-Einstein condensates.

We construct higher-order rogue wave solutions and breather profiles for the quasi-one-dimensional Gross-Pitaevskii equation with a time-dependent interatomic interaction and external trap through the similarity transformation technique. We consider three different forms of traps: (i) the time-independent expulsive trap, (ii) time-dependent monotonous trap, and (iii) time-dependent periodic trap. Our results show that when we change a parameter appearing in the time-independent or time-dependent trap the second- and third-order rogue waves transform into the first-order-like rogue waves. We also analyze the density profiles of breather solutions. Here we also show that the shapes of the breathers change when we tune the strength of the trap parameter. Our results may help to manage rogue waves experimentally in a BEC system.

Darboux transformation and Rogue waves of the Kundu-nonlinear Schrödinger equation

In this paper, the Darboux transformation of the Kundu–nonlinear Schrödinger equation is derived and generalized to the matrix of n-fold Darboux transformation. From known solution Q, the determinant representation of n-th new solutions of Q[n] are obtained by the n-fold Darboux transformation. Then soliton solutions and positon solutions are generated from trivial seed solutions, breather solutions and rogue wave solutions that are obtained from periodic seed solutions. After that, the higher order rogue wave solutions of the Kundu–nonlinear Schrödinger equation are given. We show that free parameters in eigenfunctions can adjust the patterns of the higher order rogue waves. Meanwhile, the third-order rogue waves are given explicitly. Copyright © 2014 John Wiley & Sons, Ltd.

Few-cycle optical rogue waves: complex modified Korteweg-de Vries equation.

In this paper, we consider the complex modified Korteweg-de Vries (mKdV) equation as a model of few-cycle optical pulses. Using the Lax pair, we construct a generalized Darboux transformation and systematically generate the first-, second-, and third-order rogue wave solutions and analyze the nature of evolution of higher-order rogue waves in detail. Based on detailed numerical and analytical investigations, we classify the higher-order rogue waves with respect to their intrinsic structure, namely, fundamental pattern, triangular pattern, and ring pattern. We also present several new patterns of the rogue wave according to the standard and nonstandard decomposition. The results of this paper explain the generalization of higher-order rogue waves in terms of rational solutions. We apply the contour line method to obtain the analytical formulas of the length and width of the first-order rogue wave of the complex mKdV and the nonlinear Schrödinger equations. In nonlinear optics, the higher-order rogue wave solutions found here will be very useful to generate high-power few-cycle optical pulses which will be applicable in the area of ultrashort pulse technology.

Nth-order rogue wave solutions of the complex modified Korteweg–de Vries equation

In this paper, a generalized Darboux transformation for the complex modified Korteweg–de Vries (mKdV) equation is constructed by the Darboux matrix method. As applications, Nth-order wave solutions of the complex mKdV equation have been obtained, part of which exhibit features of rogue waves, such as the first- to third-order rogue waves that are discussed and illustrated through some figures.