深水重力波紧致方程的特解
Special solutions to a compact equation for deep-water gravity waves
AI总结：
本文利用Petviashvili方法数值构造了深水重力波紧致方程的孤立波解，研究了其稳定性，发现了具有楔形奇点的奇异行波（peakons），并通过扰动方程解析得到了指数型peakons，最后利用高精度谱方法发现光滑孤立波呈弹性碰撞，暗示了Zakharov方程的可积性。
摘要：
最近，Dyachenko & Zakharov (2011) 推导了著名的Zakharov积分微分方程的一种紧致形式，该方程描述了具有自由表面的不可压缩、无限深流体势流的三阶哈密顿动力学。利用Petviashvili方法数值构造了该紧致方程的特殊行波解，并研究了它们的稳定性。此外，数值发现了具有楔形奇点的不稳定行波，即peakons。为了深入了解奇异行波的性质，我们考虑了紧致方程的扰动版本这一学术案例，并推导了具有指数形状的解析peakons。最后，通过精确的傅里叶型谱方案发现，光滑孤立波似乎发生弹性碰撞，这表明Zakharov方程具有可积性。
英文摘要：
Recently, Dyachenko & Zakharov (2011) have derived a compact form of the well known Zakharov integro-differential equation for the third order Hamiltonian dynamics of a potential flow of an incompressible, infinitely deep fluid with a free surface. Special traveling wave solutions of this compact equation are numerically constructed using the Petviashvili method. Their stability properties are also investigated. Further, unstable traveling waves with wedge-type singularities, viz. peakons, are numerically discovered. To gain insights into the properties of singular traveling waves, we consider the academic case of a perturbed version of the compact equation, for which analytical peakons with exponential shape are derived. Finally, by means of an accurate Fourier-type spectral scheme it is found that smooth solitary waves appear to collide elastically, suggesting the integrability of the Zakharov equation.
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