Recent developments of extreme wave events in integrable resonant systems

Pan Chang-Chang; Baronio Fabio; Chen Shi-Hua

doi:10.7498/aps.69.20191240

Abstract

From a microscopic perspective, the single extreme rogue wave event can be thought of as the spatiotemporally localized rational solutions of the underlying integrable model. A typical example is the fundamental Peregrine rogue wave, who in general entails a three-fold peak amplitude, while making its peak position arbitrary on a finite continuous-wave background. This kind of bizarre wave structure agrees well with the fleeting nature of realistic rogue waves and has been confirmed experimentally, first in nonlinear fibers, then in water wave tanks and plasmas, and recently in an irregular oceanic sea state. In this review, with a brief overview of the current state of the art of the concepts, methods, and research trends related to rogue wave events, we mainly discuss the fundamental Peregrine rogue wave solutions as well as their recent progress, intended for three typical integrable models, namely, the long-wave short-wave resonant equation, the three-wave resonant interaction equation, and the nonlinear Schrödinger and Maxwell–Bloch equation. Basically, while the first two models can describe the resonant interaction among optical waves, the latter governs the interaction between the optical waves and the resonant medium. For each integrable model, we present explicitly its Lax pair, Darboux transformation formulas, and fundamental Peregrine rogue wave solutions, in a self-consistent way. We confirm by convincing examples that these fundamental rogue wave solutions exhibit universality and can be applied to the multi-component or the higher-order versions of the current integrable models. By means of numerical simulations, we demonstrate as well several novel rogue wave dynamics such as coexisting rogue waves, complementary rogue waves, and Peregrine solitons of self-induced transparency.

Keywords:

Authors and contacts

1.
School of Physics, Southeast University, Nanjing 211189, China
2.
Dipartimento di Ingegneria dell’Informazione, Università di Brescia, Via Branze 38, 25123 Brescia, Italy

Corresponding author: Chen Shi-Hua, cshua@seu.edu.cn

Funds: Project supported by the National Natural Science Foundation of China (Grant Nos. 11474051, 11974075)

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Cited By

图 1 数值模拟验证初始白噪声微扰下的基阶RW解(5)式, (6)式和 (10)式的稳定性, 左列图对应$(m, n) =(-1.3514, ~0.7803)$, 中列图对应$(m, n) =(- 0.4287, ~0.6442)$. 右列图显示这两类RW结构在同一背景场中的数值激发. 图改编自文献[109]

Figure 1. Simulations confirm the stability of the fundamental RW solutions (5), (6), and (10) against initial white noise perturbations. Left column: $(m, n) =(-1.3514, ~0.7803)$; Middle column: $(m, n) =(- 0.4287, ~0.6442)$. The right column shows the numerical excitation of such two rogue wave families from the same background field. Figure adapted from Ref. [109].

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图 2 互补型基阶RW解(18)式的数值模拟结果. 左列图: 未微扰情形; 右列图: 白噪声微扰情形. 图摘自文献[122]

Figure 2. Simulation results of the complementary fundamental rogue wave solutions (18). Left column: unperturbed; Right column: perturbed by initial white noises. Figure adapted from Ref. [122].

DownLoad: Full-Size Img PowerPoint

图 3 NLS–MB方程的基阶RW解(23)的时空演化, 其中(a)列图对应解析解的3D曲面和轮廓图; (b)列图为数值模拟结果, 初始条件已文中给出; (c)列图显示这类异常波结构在背景场中的数值激发产生, 已黑线圈出. 图改编自文献[95]

Figure 3. Spatiotemporal evolution of the fundamental rogue wave solutions (23) of the NLS–MB equation. Column (a): Analytical solutions, given by 3D surface and contour plots; Column (b) the numerical results, with initial conditions being specified in the text; The column (c) shows the numerical excitation of the rogue waves, indicated by the black circles, from the background field. Figure adapted from Ref. [95].

DownLoad: Full-Size Img PowerPoint

map

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