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Nonlinear Wave and Solitons
Published in K.T. Chau, Applications of Differential Equations in Engineering and Mechanics, 2019
Figure 9.13plots the Peregrine breather profile as a function of time. The soliton-like solution oscillates with time and decays to zero as t → ∞. Figure 9.14 plots the real part, the imaginary part, and the absolute magnitude of the Peregrine breather versus time t and spatial distance x. A very distinct peak resembling a rogue wave with one hole in front and one hole behind was observed in Figure 9.14. A rogue wave is normally referred as a single wave with an unusually large magnitude, which is also known as a freak wave.
A novel method for solving third-order nonlinear Schrödinger equation by deep learning
Published in Waves in Random and Complex Media, 2022
In recent years, rogue waves (also known as freak waves, extreme waves, huge waves, etc.) have been observed in the fields of ocean [6], finance [7], nonlinear optics [8], and so on. According to the research, rogue waves occur in the midocean and seacoast. Rogue waves are huge waves limited to time and space, and are likely to pose a danger to large ships and nuclear structures located on the coast due to their without warning appearance characteristics. This phenomenon has attracted wide attention in the academic world, and many scholars have studied rogue waves. Bala and Gill [9] studied the solitons and rogue waves of Kadomstev–Petviashvili equation. Zhang et al. [10] discussed lump solutions, classical lump solutions, and rogue waves of the (2+1)-dimensional Caudrey–Dodd–Gibbon–Kotera–Sawada-like equation in 2021. Du et al. [11] studied lumps and rogue waves on the periodic backgrounds of a (2 + 1)-dimensional nonlinear Schrödinger equation in 2021. Lan [12] and Su studied solitary and rogue waves of the non-autonomous generalized AB system in 2019. Although some achievements have been made in understanding this natural phenomenon, the calculation of rogue waves in partial differential equations (PDEs) is complicated and difficult.
Dust-ion acoustic rogue waves in six-component dusty plasma
Published in Waves in Random and Complex Media, 2022
Abeer A. Mahmoud, Essam M. Abulwafa, Ahmed M. Bedeir, Atalla M. Elhanbaly
Rogue waves are the most fascinating destructive waves in nature that appear from nowhere and disappear without a trace [1–3]. Besides the plasma subject, rogue waves have been observed in different fields. In the ocean, rogue waves are naturally generated due to the instability of random conditions that tend to make the value of central amplitudes much higher than that of the wave crests around them [4,5]. Also, rogue waves can be experientially examined in some optical systems when the optical radiations propagate in photonic crystal fibers [6,7]. However, the formation of the rogue waves is primarily due to modulational instability, which occurs when a slight external perturbation of a plane wave causes such instability and in turn produces a wave with high amplitude. Various mathematical techniques have been developed to understand the occurrence of rogue waves. In contrast to the linear theories, the nonlinear analysis can be only used to deal with the dramatic behavior of rogue waves. Nonlinear Schrödinger equation (NLSE) is one of the most important equations used to explain the occurrence of such waves. This equation (NLSE) can be constructed from the plasma fluid dynamical model by employing multi-scales perturbation theory in both space and time [8,9]. Based on the nonlinear Schrödinger equation, the nonlinear structure of the rogue waves has been analytically and numerically investigated [3,10,11].
Vector semirational rogue waves for the coupled nonlinear Schrödinger equations with the higher-order effects in the elliptically birefringent optical fiber
Published in Waves in Random and Complex Media, 2020
Cui-Cui Ding, Yi-Tian Gao, Jing-Jing Su, Gao-Fu Deng, Shu-Liang Jia
Rogue waves have been seen as the large-amplitude waves unexpectedly appearing in the ocean which are capable of having disastrous effects on ships [1]. Multiple observations have confirmed the existence of rogue waves [1–11]. Analyzing the vector semirational rogue waves for the coupled nonlinear Schrödinger (NLS) equations, people have found that the interaction between the dark–bright wave and the Peregrine rogue wave may be interpreted as a generation mechanism of one rogue wave out of a slowly moving boomeronic soliton [12]. Even though the rogue waves have been first observed in the ocean [1], investigations have shown that they also arise in optical fibers [2], Bose–Einstein condensates [4], atmosphere situations [6], fluids [7,8], proteins [9] and plasmas [10]. Peregrine soliton, a first-order rational solution of the NLS equation, has been viewed as one type of the mathematical description of a rogue wave [13]. Experiments have provided a path to generate the Peregrine solitons in optical fibers [14]. Peregrine solitons have also been observed in the water-wave tanks [15] and multi-component plasmas [16].