Institut FEMTO-ST, UMR 6174 CNRS-Université de Franche-Comté, Besançon, France.
School of Mathematical Sciences, University College Dublin, Belfield, Dublin 4, Ireland.
Sci Rep. 2015 May 20;5:10380. doi: 10.1038/srep10380.
The nonlinear Schrödinger equation (NLSE) is a seminal equation of nonlinear physics describing wave packet evolution in weakly-nonlinear dispersive media. The NLSE is especially important in understanding how high amplitude "rogue waves" emerge from noise through the process of modulation instability (MI) whereby a perturbation on an initial plane wave can evolve into strongly-localised "breather" or "soliton on finite background (SFB)" structures. Although there has been much study of such structures excited under controlled conditions, there remains the open question of how closely the analytic solutions of the NLSE actually model localised structures emerging in noise-seeded MI. We address this question here using numerical simulations to compare the properties of a large ensemble of emergent peaks in noise-seeded MI with the known analytic solutions of the NLSE. Our results show that both elementary breather and higher-order SFB structures are observed in chaotic MI, with the characteristics of the noise-induced peaks clustering closely around analytic NLSE predictions. A significant conclusion of our work is to suggest that the widely-held view that the Peregrine soliton forms a rogue wave prototype must be revisited. Rather, we confirm earlier suggestions that NLSE rogue waves are most appropriately identified as collisions between elementary SFB solutions.
非线性薛定谔方程（NLSE）是描述弱非线性弥散介质中波包演化的非线性物理学的重要方程。NLSE 在理解如何通过调制不稳定性（MI）过程从噪声中产生高幅度“异常波”方面尤为重要，其中初始平面波上的微扰可以演变成强局部化的“呼吸子”或“有限背景上的孤子（SFB）”结构。尽管已经对受控制条件下激发的这种结构进行了大量研究，但 NLSE 的解析解实际上如何模拟噪声激发的 MI 中出现的局部化结构仍然是一个悬而未决的问题。我们在这里使用数值模拟来解决这个问题，将大量噪声激发 MI 中出现的峰的特性与 NLSE 的已知解析解进行比较。我们的结果表明，混沌 MI 中观察到了基本呼吸子和高阶 SFB 结构，噪声诱导峰的特征紧密围绕着 NLSE 的解析预测聚类。我们工作的一个重要结论是，建议重新审视关于 Peregrine 孤子形成异常波原型的普遍观点。相反，我们证实了早先的建议，即 NLSE 异常波最适合被识别为基本 SFB 解之间的碰撞。
