论文标题
非线性本杰明·弗弗的不稳定性 - 汉密尔顿动力学，原始呼吸器和稳定的解决方案
The nonlinear Benjamin-Feir instability -- Hamiltonian dynamics, primitive breathers, and steady solutions
论文作者
论文摘要
我们开发了一个通用框架，以描述深水重力波单向退化四重奏的立方非线性相互作用。从离散的Zakharov方程开始，因此在光谱带宽的情况下，我们根据动态相和模态振幅得出了平面汉密尔顿系统。这是两个自由参数的特征：波动作用和载波和侧带之间的模式分离。模式分离用作分叉参数，这使我们能够完全对动态进行分类。我们系统的中心对应于非平凡的，稳态的几乎共鸣的归化四重奏。马鞍点的存在与均匀和双重波列的不稳定性有关，从而概括了本杰明·弗诺（Benjamin-Feir）不稳定性的经典图片。此外，发现杂斜轨道对应于原始的三模式呼吸溶液，包括著名的非线性Schrödinger方程的Akhmediev呼吸溶液的类似物。
We develop a general framework to describe the cubically nonlinear interaction of a unidirectional degenerate quartet of deep-water gravity waves. Starting from the discretised Zakharov equation, and thus without restriction on spectral bandwidth, we derive a planar Hamiltonian system in terms of the dynamic phase and a modal amplitude. This is characterised by two free parameters: the wave action and the mode separation between the carrier and the side-bands. The mode separation serves as a bifurcation parameter, which allows us to fully classify the dynamics. Centres of our system correspond to non-trivial, steady-state nearly-resonant degenerate quartets. The existence of saddle-points is connected to the instability of uniform and bichromatic wave trains, generalising the classical picture of the Benjamin-Feir instability. Moreover, heteroclinic orbits are found to correspond to primitive, three-mode breather solutions, including an analogue of the famed Akhmediev breather solution of the nonlinear Schrödinger equation.
