Understanding the phenomenon of rogue wave formation, often called extreme waves, in diverse branches of nonlinear science has become one of the most attractive domains. Given the great richness of the new results and the increasing number of disciplines involved, we are focusing here on two pioneering fields: hydrodynamics and nonlinear optics. This tutorial aims to provide basic background and the recent developments on the formation of rogue waves in various systems in nonlinear optics, including laser physics and fiber optics. For this purpose we first discuss their formation in conservative systems, because most of the theoretical and analytical results have been realized in this context. By using a multiple space–time scale analysis, we review the derivation of the nonlinear Schrödinger equation from Maxwell’s equations supplemented by constitutive equations for Kerr materials. This fundamental equation describes the evolution of a slowly varying envelope of dispersive waves. This approximation has been widely used in the majority of systems, including plasma physics, fluid mechanics, and nonlinear fiber optics. The basic property of this generic model that governs the dynamics of many conservative systems is its integrability. In particular, we concentrate on a nonlinear regime where classical prototypes of rogue wave solutions, such as Akhmediev breathers, Peregrine, and Ma solitons are discussed as well as their experimental evidence in optics and hydrodynamics. The second part focuses on the generation of rogue waves in one- and two-dimensional dissipative optical systems. Specifically, we consider Kerr-based resonators for which we present a detailed derivation of the Lugiato–Lefever equation, assuming that the resonator length is shorter than the space scales of diffraction (or the time scale of the dispersion) and the nonlinearity. In addition, the system possesses a large Fresnel number, i.e., a large aspect ratio so that the resonator boundary conditions do not alter the central part of the beam. Dissipative structures such as solitons and modulational instability and their relation to frequency comb generation are discussed. The formation of rogue waves and the control employing time-delayed feedback are presented for both Kerr and semiconductor-based devices. The last part presents future perspectives on rogue waves to three-dimensional dispersive and diffractive nonlinear resonators.

中文翻译：

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非线性光学中的流氓波

了解非线性科学的不同分支中的流氓波形成现象（通常称为极端波）已成为最具吸引力的领域之一。鉴于新成果的丰富性和所涉及的学科数量不断增加，我们将重点关注两个开创性领域：流体动力学和非线性光学。本教程旨在提供有关非线性光学（包括激光物理学和光纤光学）中各种系统中流氓波形成的基本背景和最新进展。为此，我们首先讨论它们在保守系统中的形成，因为大多数理论和分析结果都是在这种情况下实现的。通过使用多时空尺度分析，我们回顾了从麦克斯韦方程得到非线性薛定谔方程的推导，并辅以克尔材料的本构方程。这个基本方程描述了色散波的缓慢变化包络的演变。这种近似已广泛用于大多数系统，包括等离子体物理、流体力学和非线性光纤。这个控制许多保守系统动力学的通用模型的基本属性是它的可积性。特别是，我们专注于非线性方案，其中讨论了流氓波解决方案的经典原型，例如 Akhmediev 呼吸器、Peregrine 和 Ma 孤子，以及它们在光学和流体动力学中的实验证据。第二部分侧重于一维和二维耗散光学系统中流氓波的产生。具体来说，我们考虑基于 Kerr 的谐振器，我们对此给出了 Lugiato-Lefever 方程的详细推导，假设谐振器长度短于衍射的空间尺度（或色散的时间尺度）和非线性。此外，该系统具有较大的菲涅耳数，即较大的纵横比，因此谐振器边界条件不会改变光束的中心部分。讨论了诸如孤子和调制不稳定性等耗散结构及其与频率梳产生的关系。对于 Kerr 和基于半导体的设备，提出了流氓波的形成和采用延时反馈的控制。最后一部分介绍了对三维色散和衍射非线性谐振器的恶意波的未来展望。我们考虑基于 Kerr 的谐振器，我们给出了 Lugiato-Lefever 方程的详细推导，假设谐振器长度小于衍射的空间尺度（或色散的时间尺度）和非线性。此外，该系统具有较大的菲涅耳数，即较大的纵横比，因此谐振器边界条件不会改变光束的中心部分。讨论了诸如孤子和调制不稳定性等耗散结构及其与频率梳产生的关系。对于 Kerr 和基于半导体的设备，提出了流氓波的形成和采用延时反馈的控制。最后一部分介绍了对三维色散和衍射非线性谐振器的恶意波的未来展望。我们考虑基于 Kerr 的谐振器，我们给出了 Lugiato-Lefever 方程的详细推导，假设谐振器长度小于衍射的空间尺度（或色散的时间尺度）和非线性。此外，该系统具有较大的菲涅耳数，即较大的纵横比，因此谐振器边界条件不会改变光束的中心部分。讨论了诸如孤子和调制不稳定性等耗散结构及其与频率梳产生的关系。对于 Kerr 和基于半导体的设备，提出了流氓波的形成和采用延时反馈的控制。最后一部分介绍了对三维色散和衍射非线性谐振器的恶意波的未来展望。