Wen Xiao-Yong, Yan Zhenya
Key Laboratory of Mathematics Mechanization, Institute of Systems Science, AMSS, Chinese Academy of Sciences, Beijing 100190, China.
Chaos. 2015 Dec;25(12):123115. doi: 10.1063/1.4937925.
We study higher-order rogue wave (RW) solutions of the coupled integrable dispersive AB system (also called Pedlosky system), which describes the evolution of wave-packets in a marginally stable or unstable baroclinic shear flow in geophysical fluids. We propose its continuous-wave (CW) solutions and existent conditions for their modulation instability to form the rogue waves. A new generalized N-fold Darboux transformation (DT) is proposed in terms of the Taylor series expansion for the spectral parameter in the Darboux matrix and its limit procedure and applied to the CW solutions to generate multi-rogue wave solutions of the coupled AB system, which satisfy the general compatibility condition. The dynamical behaviors of these higher-order rogue wave solutions demonstrate both strong and weak interactions by modulating parameters, in which some weak interactions can generate the abundant triangle, pentagon structures, etc. Particularly, the trajectories of motion of peaks and depressions of profiles of the first-order RWs are explicitly analyzed. The generalized DT method used in this paper can be extended to other nonlinear integrable systems. These results may be useful for understanding the corresponding rogue-wave phenomena in fluid mechanics and related fields.
我们研究了耦合可积色散AB系统（也称为佩德洛斯基系统）的高阶 rogue 波（RW）解，该系统描述了地球物理流体中处于边际稳定或不稳定斜压切变流中波包的演化。我们给出了其连续波（CW）解及其调制不稳定性形成 rogue 波的存在条件。根据达布矩阵中谱参数的泰勒级数展开及其极限过程，提出了一种新的广义N重达布变换（DT），并将其应用于CW解以生成耦合AB系统的多 rogue 波解，这些解满足一般相容性条件。这些高阶 rogue 波解的动力学行为通过调节参数展示了强相互作用和弱相互作用，其中一些弱相互作用可以产生丰富的三角形、五边形结构等。特别地，明确分析了一阶RWs 轮廓的峰值和谷值的运动轨迹。本文中使用的广义DT方法可以扩展到其他非线性可积系统。这些结果可能有助于理解流体力学及相关领域中相应的 rogue 波现象。
