Wu C F, Grimshaw R H J, Chow K W, Chan H N
Department of Mathematics, University of Hong Kong, Pokfulam, Hong Kong.
Department of Mathematics, University College London, London WC1E 6BT, United Kingdom.
Chaos. 2015 Oct;25(10):103113. doi: 10.1063/1.4931708.
Rogue waves are unexpectedly large and localized displacements from an equilibrium position or an otherwise calm background. For the nonlinear Schrödinger (NLS) model widely used in fluid mechanics and optics, these waves can occur only when dispersion and nonlinearity are of the same sign, a regime of modulation instability. For coupled NLS equations, rogue waves will arise even if dispersion and nonlinearity are of opposite signs in each component as new regimes of modulation instability will appear in the coupled system. The same phenomenon will be demonstrated here for a coupled "AB" system, a wave-current interaction model describing baroclinic instability processes in geophysical flows. Indeed, the onset of modulation instability correlates precisely with the existence criterion for rogue waves for this system. Transitions from "elevation" rogue waves to "depression" rogue waves are elucidated analytically. The dispersion relation as a polynomial of the fourth order may possess double pairs of complex roots, leading to multiple configurations of rogue waves for a given set of input parameters. For special parameter regimes, the dispersion relation reduces to a cubic polynomial, allowing the existence criterion for rogue waves to be computed explicitly. Numerical tests correlating modulation instability and evolution of rogue waves were conducted.
rogue波是相对于平衡位置或平静背景出现的意外的大幅局部位移。对于流体力学和光学中广泛使用的非线性薛定谔（NLS）模型，这些波仅在色散和非线性具有相同符号时才会出现，即调制不稳定性状态。对于耦合的NLS方程，即使每个分量中的色散和非线性具有相反的符号，也会出现rogue波，因为耦合系统中会出现新的调制不稳定性状态。本文将针对耦合的“AB”系统展示相同的现象，该系统是一个描述地球物理流中斜压不稳定过程的波流相互作用模型。实际上，调制不稳定性的起始与该系统中rogue波的存在准则精确相关。从“隆起”rogue波到“凹陷”rogue波的转变通过解析得到阐明。作为四阶多项式的色散关系可能具有两对复根，导致对于给定的一组输入参数存在多种rogue波配置。对于特殊的参数范围，色散关系简化为三次多项式，从而可以明确计算rogue波的存在准则。进行了将调制不稳定性与rogue波演化相关联的数值测试。
