广义Green-Naghdi方程的Hamilton形式
Hamiltonian formulation of the extended Green-Naghdi equations
摘要：
本文提出了一种新的方法，将格林-纳格迪（GN）浅水模型方程推广到包含任意高阶色散效应的一般系统。作为一个说明性的例子，我们推导出一个模型方程，它是精确到四次幂的浅度参数，同时保持GN方程的完全非线性，并获得其孤立波解的奇异摄动分析。我们证明了推广的GN方程具有与GN方程相同的哈密顿结构。我们还证明了Zakharov的表面重力波的哈密顿公式是等效的扩展GN系统重写前系统的动量密度，而不是在自由表面的速度势。
英文摘要：
A novel method is developed for extending the Green-Naghdi (GN) shallow-water model equation to the general system which incorporates the arbitrary higher-order dispersive effects. As an illustrative example, we derive a model equation which is accurate to the fourth power of the shallowness parameter while preserving the full nonlinearity of the GN equation, and obtain its solitary wave solutions by means of a singular perturbation analysis. We show that the extended GN equations have the same Hamiltonian structure as that of the GN equation. We also demonstrate that Zakharov's Hamiltonian formulation of surface gravity waves is equivalent to that of the extended GN system by rewriting the former system in terms of the momentum density instead of the velocity potential at the free surface.
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