Coppini F, Grinevich P G, Santini P M
PhD Program in Physics, Dipartimento di Fisica, Università di Roma "La Sapienza," and Istituto Nazionale di Fisica Nucleare (INFN), Sezione di Roma, Piazzale Aldo Moro 2, I-00185 Roma, Italy.
Steklov Mathematical Institute of Russian Academy of Sciences, 8 Gubkina Street, Moscow, 199911, Russia and L. D. Landau Institute for Theoretical Physics, Prospekt Akademika Semenova 1a, Chernogolovka 142432, Russia.
Phys Rev E. 2020 Mar;101(3-1):032204. doi: 10.1103/PhysRevE.101.032204.
The focusing nonlinear Schrödinger (NLS) equation is the simplest universal model describing the modulation instability of quasimonochromatic waves in weakly nonlinear media, the main physical mechanism for the appearance of anomalous (rogue) waves (AWs) in nature. In this paper, concentrating on the simplest case of a single unstable mode, we study the special Cauchy problem for the NLS equation perturbed by a linear loss or gain term, corresponding to periodic initial perturbations of the unstable background solution of the NLS. Using the finite gap method and the theory of perturbations of soliton partial differential equations, we construct the proper analytic model describing quantitatively how the solution evolves after a suitable transient into slowly varying lower dimensional patterns (attractors) on the (x,t) plane, characterized by ΔX=L/2 in the case of loss and by ΔX=0 in the case of gain, where ΔX is the x shift of the position of the AW during the recurrence, and L is the period. This process is described, to leading order, in terms of elementary functions of the initial data. Since dissipation can hardly be avoided in all natural phenomena involving AWs, and since a small dissipation induces O(1) effects on the periodic AW dynamics, generating the slowly varying loss or gain attractors analytically described in this paper, we expect that these attractors together with their generalizations corresponding to more unstable modes will play a basic role in the theory of periodic AWs in nature.
聚焦非线性薛定谔（NLS）方程是描述弱非线性介质中准单色波调制不稳定性的最简单通用模型，这是自然界中出现异常（ rogue ）波（AWs）的主要物理机制。在本文中，专注于单个不稳定模式的最简单情况，我们研究了由线性损耗或增益项扰动的NLS方程的特殊柯西问题，这对应于NLS不稳定背景解的周期性初始扰动。利用有限隙方法和孤子偏微分方程的微扰理论，我们构建了合适的解析模型，定量描述了在适当的瞬态之后，解如何演化为（x,t）平面上缓慢变化的低维模式（吸引子），在损耗情况下其特征为ΔX = L/2，在增益情况下其特征为ΔX = 0，其中ΔX是AW在复发期间位置的x位移，L是周期。这个过程在初始数据的初等函数方面被描述到主导阶。由于在涉及AW的所有自然现象中几乎都不可避免地存在耗散，并且由于小的耗散会对周期性AW动力学产生O(1)效应，从而产生本文中解析描述的缓慢变化的损耗或增益吸引子，我们预计这些吸引子及其对应于更不稳定模式的推广将在自然界中周期性AW的理论中发挥基本作用。
