Higher-order rogue waves in the nonlocal defocusing coupled nonlinear Schrödinger equation

The previous studies have shown that the defocusing NLSE has no the modulational instability, and was not found to admit the rogue wave phenomenon so far. In this paper, we address the question of the higher-order rogue wave solutions of the nonlocal $\mathscr{P}\mathscr{T}$-symmetric NLSE in the defocusing regime. Based on Darboux transformation and iterations, we derive an explicit solution for the higher-order rogue waves by adopting a variable separation and Taylor expansion technique. The higher-order rogue wave solutions are expressed in separation-of-variables form. Furthermore, in order to understand these solutions better, patterns of the rogue waves for lowest three order are explored clearly and conveniently. The reported results may be useful for the design of experiments for observation of rogue waves in the defocusing nonlinear physical systems.

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.

Breather interactions and higher-order nonautonomous rogue waves for the inhomogeneous nonlinear Schrödinger Maxwell–Bloch equations

General line rogue waves in the Mel’nikov equation are derived via the Hirota bilinear method, which are given in terms of determinants whose matrix elements have plain algebraic expressions. It is shown that fundamental rogue waves are line rogue waves, which arise from the constant background with a line profile and then disappear into the constant background again. By means of the regulation of free parameters, two subclass of nonfundamental rogue waves are generated, which are called as multirogue waves and higher-order rogue waves. The multirogue waves consist of several fundamental line rogue waves, which arise from the constant background and then decay back to the constant background. The higher-order rogue waves start from a localised lump and retreat back to it. The dynamical behaviours of these line rogue waves are demonstrated by the density and the three-dimensional figures.

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.

This paper studies the multi-component derivative nonlinear Schrödinger (n-DNLS) equations featuring nonzero boundary conditions. Employing the Darboux transformation method, we derive higher-order vector rogue wave solutions for the n-DNLS equations. Specifically, we focus on the distinctive scenario where the (n+1)-order characteristic polynomial possesses an explicit (n+1)-multiple root. Additionally, we provide an in-depth analysis of the asymptotic dynamic behaviors and pattern classification inherent to the higher-order vector rogue wave solution of the n-DNLS equations, mainly when one of the internal arbitrary parameters is extremely large. These patterns are related to the root structures in the generalized Wronskian-Hermite polynomial hierarchies.

In this paper, the (3+1)-dimensional generalized B-type Kadomtsev–Petviashvili equation for water waves is investigated. Through the Hirota method and Kadomtsev–Petviashvili hierarchy reduction, we obtain the first-order, higher-order, multiple rogue waves and lump solitons based on the solutions in terms of the Gramian. The first-order rogue waves are the line rogue waves which arise from the constant background and then disappear into the constant background again, while the first-order lump solitons propagate stably. Interactions among several first-order rogue waves which are described by the multiple rogue waves are presented. Elastic interactions of several first-order lump solitons are also presented. We find that the higher-order rogue waves and lump solitons can be treated as the superpositions of several first-order ones, while the interaction between the second-order lump solitons is inelastic.

In this paper, we investigate a third-order nonlinear Schrödinger equation (NLSE) which is used to describe the propagation of ultrashort pulses in the subpicosecond or femtosecond regime. Based on the independent transformation, the bilinear form of the third-order NLSE is constructed. The multiple soliton solutions are constructed by solving the bilinear form. The higher-order rogue waves and interaction between one-soliton and first-order rogue wave are obtained by the long wave limit in multi-solitons. The dynamics of the first-order rogue wave, second-order rogue wave and interaction between one-soliton and first-order rogue wave are presented by selecting the appropriate parameters. In particular parameters, the positions and the maximum of amplitude of rogue wave can be confirmed by the detailed calculations.

Higher-order rogue wave solutions for a generalized Hirota-Maxwell-Bloch system in the erbium-doped fibers

In this work, we study a prototypical, experimentally accessible scenario that enables the systematic generation of so-called high-order rogue waves in atomic Bose-Einstein condensates. These waveforms lead to significantly and controllably more extreme focusing events than the famous Peregrine soliton. In one spatial dimension, we showcase conclusive numerical evidence that our scheme generates the focusing behavior associated with the first four rogue waves from the relevant hierarchy. We then extend considerations to anisotropic two-dimensional and even three-dimensional settings, establishing that the scheme can generate second-order rogue waves despite the well-known limitation of finite-time blow up of focusing nonlinear Schrödinger equations.

The long-wave–short-wave resonance interaction model: Generalized higher-order semi-rational rogue wave solutions and their dynamics

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.

General rogue wave solutions and their dynamics in the complex modified Korteweg–de Vries equation

We employ a nonrecursive Darboux transformation formalism for obtaining a hierarchy of rogue wave solutions to the focusing vector nonlinear Schrödinger equations (Manakov system). The exact explicit rogue wave solutions up to the second order are presented. Typical rogue wave patterns such as Peregrine-type, triple, quadruple, and sextuple vector rogue waves, either bright–dark or bright–bright in their respective components, are put forward. Despite the diversity, there exists a universal compossibility that different rogue wave states could coexist for the same background parameters. It is also shown that the higher-order rogue wave hierarchy can indeed be thought of as a nonlinear superposition of a fixed well prescribed number of fundamental rogue waves. These results may help understand the protean rogue wave manifestations in areas ranging from hydrodynamics to nonlinear optics.

In this study, we present the rigorous theory of the robust inverse scattering method for the discrete high-order nonlinear Schrödinger (HNLS) equation with a nonzero boundary condition (NZBC). Using the direct scattering problem, we deduce the analyticity, symmetries, and asymptotic behaviors of the Jost solutions and scattering matrix. We also formulate the inverse scattering problem using the matrix Riemann–Hilbert problem (RHP). Furthermore, utilizing the loop group theory, we construct the multi-fold Darboux transformation (DT) within the framework of the robust inverse scattering transform. Additionally, we develop the corresponding Bäcklund transformation (BT) to obtain the multi-fold lattice soliton solutions. To derive the high-order rational solutions, we further construct the high-order DT. Finally, we theoretically and graphically analyze these solutions, which exhibit lattice breather waves, W-shape lattice solitons, high-order lattice rogue waves (RW), and their interactions.