Higher-order vector rogue waves in inhomogeneous optical media: insights into variable-coefficient coupled nonlinear schrödinger model with four-wave mixing

The N-fold Darboux transformation of the coupled Lakshmanan–Porsezian–Daniel (LPD) equations is constructed. Based on the Darboux transformation and the limiting technique, we investigate two kinds of solutions for the coupled LPD equations, which are higher-order interactional solutions and rogue wave (RW) pairs. Through considering the double-root situation of the spectral characteristic equation for the matrix in the Lax pair, we give the higher-order interactional solutions among higher-order RWs, multi-bright (dark) solitons and multi-breather. Besides, we consider the triple-root situation of the spectral characteristic equation and get the higher-order RW pairs. It demonstrates that the RW pairs are greatly different form the traditional higher-order RWs. The fist-order RW pairs can split into two traditional first-order RWs, and four or six traditional fundamental RWs can emerge from the second-order case. The corresponding dynamics of these explicit solutions are discussed in detail.

Under investigation in this paper is a variable-coefficient derivative nonlinear Schrodinger (vc-DNLS) equation governing the femtosecond pulses in the inhomogeneous optical fibers or nonlinear Alfven waves in the inhomogeneous plasmas. Higher-order breather and rogue wave solutions of the vc-DNLS equation are obtained via the variable-coefficient modified Darboux transformation. Two types of the breather interactions (the head-on and overtaking collisions) are exhibited with different spectral parameters. By suitably choosing the inhomogeneous functions, the parabolic breather, periodic breather, breather amplification and breather evolution are demonstrated. Furthermore, the characteristics of the higher-order fundamental rogue wave, periodic rogue wave and composite rogue wave are graphically discussed. Additionally, the nonlinear tunneling of the higher-order breathers and rogue waves are studied.

In this article, a fifth-order dispersive nonlinear Schrödinger equation is investigated, which describes the propagation of ultrashort optical pulses, up to the attosecond duration, in an optical fibre. Rogue wave solutions are derived by virtue of the generalised Darboux transformation. Rogue wave structures and interaction are discussed through (i) the analyses on the higher-order rogue waves, the cubic, quartic, quintic, group-velocity, and phase-parameter effects; (ii) a higher-order rogue wave consisting of the first-order rogue waves via the interaction; (iii) characteristics of the rogue waves which are summarised, including the maximum/minimum values of the rogue waves and the number of the first-order rogue waves for composing the higher-order rogue wave; and (iv) spatial-temporal patterns which are illustrated and compared with those of the ‘self-focusing’ nonlinear Schrödinger equation. We find that the quintic terms increase the time of appearance for the first-order rogue waves which form the higher-order rogue wave, and that the quintic terms affect the interaction among the first-order rogue waves, which elongates the distance of appearance for the higher-order rogue wave.

By Taylor expansion of Darboux matrix, a new generalized Darboux transformations (DTs) for a (2 + 1)-dimensional nonlinear Schrödinger (NLS) equation is derived, which can be reduced to two (1 + 1)-dimensional equation: a modified KdV equation and an NLS equation. With the help of symbolic computation, some higher-order rational solutions and rogue wave (RW) solutions are constructed by its (1, N −1)-fold DTs according to determinants. From the dynamic behavior of these rogue waves discussed under some selected parameters, we find that the RWs and solitons are demonstrated some interesting structures including the triangle, pentagon, heptagon profiles, etc. Furthermore, we find that the wave structure can be changed from the higher-order RWs into higher-order rational solitons by modulating the main free parameter. These results may give an explanation and prediction for the corresponding dynamical phenomena in some physically relevant systems.

Higher-order optical rogue waves in spatially inhomogeneous multimode fiber

The higher-order rogue wave (RW) for a spatial discrete Hirota equation is investigated by the generalized (1, )-fold Darboux transformation. We obtain the higher-order discrete RW solution to the spatial discrete Hirota equation. The fundamental RWs exhibit different amplitudes and shapes associated with the spectral parameters. The higher-order RWs display triangular patterns and pentagons with different peaks. We show the differences between the RW of the spatially discrete Hirota equation and the discrete nonlinear Schrödinger equation. Using the contour line method, we study the localization characters including the length, width, and area of the first-order RWs of the spatially discrete Hirota equation.

General higher-order semi-rational rogue wave solutions within one-dimensional three-wave resonant interaction systems are investigated, utilizing the bilinear Kadomtsev-Petviashvili hierarchy reduction method. These solutions, formulated as determinants involving Schur polynomials and exponential functions, are classified into three distinct categories: (i) N1-th order rogue waves with one simple root, accompanied by N2-th order breathers and N3-th order dark solitons as coexisting backgrounds; (ii) (Ñ1,Ñ2)-th order rogue waves with two simple roots, accompanied by N2-th order breathers as backgrounds; and (iii) (Ñ1,Ñ2)-th order rogue waves with one double root, accompanied by N2-th order breathers as backgrounds. The results demonstrate that pure rogue waves, pure breathers, pure dark solitons, and their various combinations can be derived from these general higher-order semi-rational rogue wave solutions by selecting appropriate nonzero integers N1, N2, N3, Ñ1 and Ñ2. A significant advancement is the introduction of generalized tau functions, which improve the representation of diverse solitary wave solutions within the Kadomtsev-Petviashvili reduction framework. This methodological innovation applies to other one-dimensional integrable systems, expanding the exploration of general higher-order semi-rational rogue wave solutions. As an application, the intricate patterns and dynamic behaviors of these rogue waves, accompanied by dark solitons, breathers, and their coexistence, are analyzed and visualized through several illustrative figures.

Higher-order rational solitons and rogue-like wave solutions of the (2 + 1)-dimensional nonlinear fluid mechanics equations

We study the higher-order rogue wave solutions of the Kadomtsev Petviashvili—Benjanim Bona Mahony (KP-BBM) model with and without controllable center via the Hirota bilinear approach. To construct higher-order rogue wave solutions of the model, we apply cross-product terms in a polynomial expression as a test function. We construct three kinds of rogue waves with center at the origin and three kinds of higher-order rogue waves with a controllable center by choosing different test functions. We show how the center can control the shapes and orders of the rogue waves by choosing the values of parameters of the center. In particular, the 3-rogue wave solutions exhibit double rogue waves for lower values and distinct triple rogue waves in a triangular structure for a sufficiently large value parameters of the center. Moreover, the 6-rogue wave solutions of the model present lower-order rogue waves for small values and higher-order rogue waves for large value parameters of the center. It is shown that the order of the rogue gradually increases for rising value of parameters, and it will be maximum six distinct rogues for a sufficiently large values. Finally, we explain the solutions of the model in 3D and in density plots graphically.

The purpose of this paper is to report the feasibility of constructing high-order rogue waves with controllable fission and asymmetry for high-dimensional nonlinear evolution equations. Such a nonlinear model considered in this paper as the concrete example is the (3 + 1)-dimensional generalized Boussinesq (gB) equation, and the corresponding method is Zhaqilao’s symbolic computation approach containing two embedded parameters. It is indicated by the (3 + 1)-dimensional gB equation that the embedded parameters can not only control the center of the first-order rogue wave, but also control the number of the wave peaks split from higher-order rogue waves and the asymmetry of higher-order rogue waves about the coordinate axes. The main novelty of this paper is that the obtained results and findings can provide useful supplements to the method used and the controllability of higher-order rogue waves.

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.

By virtue of the bilinear method and the KP hierarchy reduction technique, exact explicit rational solutions of the multicomponent Mel’nikov equation and the multicomponent Schrodinger–Boussinesq equation are constructed, which contain multicomponent short waves and single-component long wave. For the multicomponent Mel’nikov equation, the fundamental rational solutions possess two different behaviours: lump and rogue wave. It is shown that the fundamental (simplest) rogue waves are line localised waves which arise from the constant background with a line profile and then disappear into the constant background again. The fundamental line rogue waves can be classified into three: bright, intermediate and dark line rogue waves. Two subclasses of non-fundamental rogue waves, i.e., multirogue waves and higher-order rogue waves are discussed. The multirogue waves describe interaction of several fundamental line rogue waves, in which interesting wave patterns appear in the intermediate time. Higher-order rogue waves exhibit dynamic behaviours that the wave structures start from lump and then retreat back to it. Moreover, by taking the parameter constraints further, general higher-order rogue wave solutions for the multicomponent Schrodinger–Boussinesq system are generated.

Rational solutions to two- and one-dimensional multicomponent Yajima–Oikawa systems

We introduce a mechanism for generating higher-order rogue waves (HRWs) of the nonlinear Schrödinger (NLS) equation: the progressive fusion and fission of n degenerate breathers associated with a critical eigenvalue λ(0) creates an order-n HRW. By adjusting the relative phase of the breathers in the interacting area, it is possible to obtain different types of HRWs. The value λ(0) is a zero point of an eigenfunction of the Lax pair of the NLS equation and it corresponds to the limit of the period of the breather tending to infinity. By employing this mechanism we prove two conjectures regarding the total number of peaks, as well as a decomposition rule in the circular pattern of an order-n HRW.

Modulation instability, higher-order rogue waves and dynamics of the Gerdjikov–Ivanov equation