Characteristics of rogue waves on a soliton background in a coupled nonlinear Schrödinger equation

In this paper, a coupled nonlinear Schrödinger (CNLS) equation, which can describe evolution of localized waves in a two‐mode nonlinear fiber, is under investigation. By using the Darboux‐dressing transformation, the new localized wave solutions of the equation are well constructed with a detailed derivation. These solutions reveal rogue waves on a soliton background. Moreover, the main characteristics of the solutions are discussed with some graphics. Our results would be of much importance in predicting and enriching rogue wave phenomena in nonlinear wave fields.     

The n‐order rogue waves of Fokas–Lenells equation

Considering certain terms of the next asymptotic order beyond the nonlinear Schrödinger equation, the Fokas–Lenells (FL) equation governed by the FL system arises as a model for nonlinear pulse propagation in optical fibers. The expressions of the q^[n] and r^[n] in the FL system are generated by the n‐fold Darboux transformation (DT), each element of the matrix is a 2 × 2 matrix, expressed by a ratio of (2n + 1) × (2n + 1) determinant and 2n × 2n determinant of eigenfunctions. Further, a Taylor series expansion about the n‐order breather solutions q^[n] generated using by DT and assuming periodic seed solutions under reduction can generate the n‐order rogue waves of the FL equation explicitly with 2n + 3 free parameters. Copyright © 2014 John Wiley & Sons, Ltd.     

Optical rogue waves for the coherently coupled nonlinear Schrödinger equation with alternate signs of nonlinearities

In this paper, optical rogue waves for the coherently coupled nonlinear Schrödinger equation with alternate signs of nonlinearities are investigated via Darboux transformation. We derive family of structures of rogue wave, including rogue waves with one peak and two valleys, bright rogue waves without valleys while with one peak or two peaks.     

Bound-state soliton and rogue wave solutions for the sixth-order nonlinear Schrödinger equation via inverse scattering transform method

In this work, inverse scattering transform for the sixth-order nonlinear Schrödinger equation with both zero and nonzero boundary conditions at infinity is given, respectively. For the case of zero boundary conditions, in terms of the Laurent's series and generalization of the residue theorem, the bound-state soliton is derived. For nonzero boundary conditions, using the robust inverse scattering transform, we present a matrix Riemann–Hilbert problem of the sixth-order nonlinear Schrödinger equation. Then, based on the obtained Riemann–Hilbert problem, the rogue wave solutions are derived through a modified Darboux transformation. Besides, according to some appropriate parameters choices, several graphical analysis are provided to discuss the dynamical behaviors of the rogue wave solutions and analyze how the higher-order terms affect the rogue wave.     

Rational and semi‐rational solutions of a nonlocal (2 + 1)‐dimensional nonlinear Schrödinger equation

We consider the fully parity‐time (PT) symmetric nonlocal (2 + 1)‐dimensional nonlinear Schrödinger (NLS) equation with respect to x and y. By using Hirota's bilinear method, we derive the N‐soliton solutions of the nonlocal NLS equation. By using the resulting N‐soliton solutions and employing long wave limit method, we derive its nonsingular rational solutions and semi‐rational solutions. The rational solutions act as the line rogue waves. The semi‐rational solutions mean different types of combinations in rogue waves, breathers, and periodic line waves. Furthermore, in order to easily understand the dynamic behaviors of the nonlocal NLS equation, we display some graphics to analyze the characteristics of these solutions.     

Generation mechanism of rogue waves for the discrete nonlinear Schrödinger equation

In this paper, we analyze the generation mechanism of rogue waves for the discrete nonlinear Schrödinger (DNLS) equation from the viewpoint of structural discontinuities. First of all, we derive the analytical breather solutions of the DNLS equation on a new nonvanishing background through the Darboux transformation (DT). Via the explicit expressions of group and phase velocities, we give the parameter conditions for existence of the velocity jumps, which are consistent with the derivation of rogue waves via the generalized DT. Furthermore, to verify such statement, we apply the Taylor expansion to the breather solutions and find that the first-order rogue wave can be obtained at the velocity-jumping point. Our analysis can help to enrich the understanding on the rogue waves for the discrete nonlinear systems.     

Novel rogue waves in an inhomogenous nonlinear medium with external potentials

Employing the similarity transformation connected with the standard constant coefficient nonlinear Schrödinger equation, we obtain the analytical rogue wave solutions to a generalized variable coefficient nonlinear Schrödinger equation with external potentials describing the pulse propagation in nonlinear media with transverse and longitudinal directions nonuniformly distributed. Based on the obtained solutions, abundant structures of rogue waves are constructed by selecting some special parameters. The main properties as well as the dynamic behaviors of these rogue waves are discussed by direct computer simulations.     

General soliton solutions to a reverse-time nonlocal nonlinear Schrödinger equation

General soliton solutions to a reverse-time nonlocal nonlinear Schrödinger (NLS) equation are discussed via a matrix version of binary Darboux transformation. With this technique, searching for solutions of the Lax pair is transferred to find vector solutions of the associated linear differential equation system. From vanishing and nonvanishing seed solutions, general vector solutions of such linear differential equation system in terms of the canonical forms of the spectral matrix can be constructed by means of triangular Toeplitz matrices. Several explicit one-soliton solutions and two-soliton solutions are provided corresponding to different forms of the spectral matrix. Furthermore, dynamics and interactions of bright solitons, degenerate solitons, breathers, rogue waves, and dark solitons are also explored graphically.     

Characteristics of rogue waves on a soliton background in the general three-component nonlinear Schrödinger equation

Under investigation in this work is the general three-component nonlinear Schrödinger equation, which is an important integrable system. The new localized wave solutions of the equation are derived using a Darboux-dressing transformation with an asymptotic expansion. These localized waves display rogue waves on a multisoliton background. Furthermore, the main characteristics of the new localized wave solutions are analyzed with some graphics. Our results indicate that more abundant and novel localized waves may exist in the multi-component coupled equations than in the uncoupled ones.     

A Robust Inverse Scattering Transform for the Focusing Nonlinear Schrödinger Equation

We propose a modification of the standard inverse scattering transform for the focusing nonlinear Schrödinger equation (also other equations by natural generalization) formulated with nonzero boundary conditions at infinity. The purpose is to deal with arbitrary-order poles and potentially severe spectral singularities in a simple and unified way. As an application, we use the modified transform to place the Peregrine solution and related higher-order “rogue wave” solutions in an inverse-scattering context for the first time. This allows one to directly study properties of these solutions such as their dynamical or structural stability, or their asymptotic behavior in the limit of high order. The modified transform method also allows rogue waves to be generated on top of other structures by elementary Darboux transformations rather than the generalized Darboux transformations in the literature or other related limit processes. © 2019 Wiley Periodicals, Inc.     

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

We construct the multi-breather solutions of the focusing nonlinear Schrödinger equation (NLSE) on the background of elliptic functions by the Darboux transformation, and express them in terms of the determinant of theta functions. The dynamics of the breathers in the presence of various kinds of backgrounds such as dn, cn, and nontrivial phase-modulating elliptic solutions are presented, and their behaviors dependent on the effect of backgrounds are elucidated. We also determine the asymptotic behaviors for the multibreather solutions with different velocities in the limit , where the solution in the neighborhood of each breather tends to the simple one-breather solution. Furthermore, we exactly solve the linearized NLSE using the squared eigenfunction and determine the unstable spectra for elliptic function background. By using them, the Akhmediev breathers arising from these modulational instabilities are plotted and their dynamics are revealed. Finally, we provide the rogue wave and higher order rogue wave solutions by taking the special limit of the breather solutions at branch points and the generalized Darboux transformation. The resulting dynamics of the rogue waves involves rich phenomena, depending on the choice of the background and possessing different velocities relative to the background. We also provide an example of the multi- and higher order rogue wave solution.     

Doubly localized two-dimensional rogue waves generated by resonant collision in Maccari system

In this paper, the Maccari system is investigated, which is viewed as a two-dimensional extension of nonlinear Schrödinger equation. We derive doubly localized two-dimensional rogue waves on the dark solitons of the Maccari system with Kadomtsev–Petviashvili hierarchy reduction method. The two-dimensional rogue waves include line segment rogue waves and rogue-lump waves, which are localized in two-dimensional space and time. These rogue waves are generated by the resonant collision of rational solitary waves and dark solitons, the whole process of transforming elastic collision into resonant collision is analytically studied. Furthermore, we also discuss the local characteristics and asymptotic properties of these rogue waves. Simultaneously, the generating conditions of the line segment rogue wave and rogue-lump wave are also given, which provides the possibility to predict rogue wave. Finally, a new way to obtain the high-order rogue waves of the nonlinear Schrödinger equation are given by proper reduction from the semi-rational solutions of the Maccari system.     

N solutions for a derivative nonlinear Schrödinger‐type equation via Riemann‐Hilbert approach

The inverse scattering transform for the derivative nonlinear Schrödinger‐type equation is studied via the Riemann‐Hilbert approach. In the direct scattering process, the spectral analysis of the Lax pair is performed, from which a Riemann‐Hilbert problem is established for the derivative nonlinear Schrödinger‐type equation. In the inverse scattering process, N‐soliton solutions of the derivative nonlinear Schrödinger‐type equation are obtained by solving Riemann‐Hilbert problems corresponding to the reflectionless cases. Moreover, the dynamics of the exact solutions are discussed.     

The realization of controllable three dimensional rogue waves in nonlinear inhomogeneous system

We obtain self-similar first-order and second-order rogue wave solutions for the (3+1)-dimensional inhomogeneous nonlinear Schrödinger equation. Based on these solutions, we investigate the control and manipulation of rogue waves in the dispersion decreasing fibers with Logarithmic profile and Gaussian profile. Our results indicate that the propagation behaviors of rogue waves, such as fast excitation, sustainment and restraint, can be manipulated by modulating the relation between the maximum value of the effective propagation distance Z_m and the parameter Z₀ relating to the excited types of rogue wave. The comparison of the propagation behavior of rogue wave in the dispersion decreasing waveguides with Logarithmic profile, Gaussian profile and hyperbolic profile is also given.     

Periodic two-phase “Rogue waves”

A family of periodic (in x and z) two-gap solutions of the focusing nonlinear Schrödinger equation is constructed. A condition under which the two-gap solutions exhibit the behavior of periodic “rogue waves” is obtained.     

The hierarchy of higher order solutions of the derivative nonlinear Schrödinger equation

In this paper, we provide a simple method to generate higher order position solutions and rogue wave solutions for the derivative nonlinear Schrödinger equation. The formulae of these higher order solutions are given in terms of determinants. The dynamics and structures of solutions generated by this method are studied.     

Breather and rogue wave solutions for the (2+1)-dimensional nonlinear Schrödinger–Maxwell–Bloch equation

In this paper, we mainly study the (2+1)-dimensional Schrödinger–Maxwell–Blochequation (SMBE). We have constructed the generalized N-fold Darboux transformations (DT), and based on the plane wave solutions, the breather and rogue wave solutions are systematically generated, the dynamical features of those solutions are graphically represented.     

On the Modulations and Nonlinear Interactions of Waves and the Nonlinear Stability of a Near-Critical System

The nonlinear interactions and modulations of an n-dimensional wave and of a disturbance to a near-critical system governed by a general (n + 1)-dimensional system of equations are studied by perturbation methods. It is found that these modulations are governed by an evolution equation which is either by itself or coupled to a second equation, depending on the nature of the long wave solutions of the corresponding linearized system. When a single evolution equation exists, its leading terms are shown to give the nonlinear Schrödinger equation. Water waves and near-critical plane Poiseuille flow are discussed as examples.     

Superposition principle and composite solutions to coupled nonlinear Schrödinger equations

We show that the superposition principle applies to coupled nonlinear Schrödinger equations with cubic nonlinearity where exact solutions may be obtained as a linear combination of other exact solutions. This is possible due to the cancelation of cross terms in the nonlinear coupling. First, we show that a composite solution, which is a linear combination of the two components of a seed solution, is another solution to the same coupled nonlinear Schrödinger equation. Then, we show that a linear combination of two composite solutions is also a solution to the same equation. With emphasis on the case of Manakov system of two-coupled nonlinear Schrödinger equations, the superposition is shown to be equivalent to a rotation operator in a two-dimensional function space with components of the seed solution being its coordinates. Repeated application of the rotation operator, starting with a specific seed solution, generates a series of composite solutions, which may be represented by a generalized solution that defines a family of composite solutions. Applying the rotation operator to almost all known exact seed solutions of the Manakov system, we obtain for each seed solution the corresponding family of composite solutions. Composite solutions turn out, in general, to possess interesting features that do not exist in the seed solution. Using symmetry reductions, we show that the method applies also to systems of N-coupled nonlinear Schrödinger equations. Specific examples for the three-coupled nonlinear Schrödinger equation are given.     

Exact traveling wave solutions to the fourth‐order dispersive nonlinear Schrödinger equation with dual‐power law nonlinearity

In this paper, we investigate exact traveling wave solutions of the fourth‐order nonlinear Schrödinger equation with dual‐power law nonlinearity through Kudryashov method and (G'/G)‐expansion method. We obtain miscellaneous traveling waves including kink, antikink, and breather solutions. These solutions may be useful in the explanation and understanding of physical behavior of the wave propagation in a highly dispersive optical medium. Copyright © 2015 John Wiley & Sons, Ltd.