This paper investigates the stability of traveling wave solutions to the free boundary Euler equations with a submerged point vortex. We prove that sufficiently small-amplitude waves with small enough vortex strength are conditionally orbitally stable. In the process of obtaining this result, we develop a quite general stability/instability theory for bound state solutions of a large class of infinite-dimensional Hamiltonian systems in the presence of symmetry. This is in the spirit of the seminal work of Grillakis, Shatah, and Strauss, but with hypotheses that are relaxed in a number of ways necessary for the point vortex system, and for other hydrodynamical applications more broadly. In particular, we are able to allow the Poisson map to have merely dense range, as opposed to being surjective, and to be state-dependent. As a second application of the general theory, we consider a family of nonlinear dispersive PDEs that includes the generalized KdV and Benjamin--Ono equations. The stability/instability of solitary waves for these systems has been studied extensively, notably by Bona, Souganidis, and Strauss, who used a modification of the GSS method. We provide a new, more direct proof of these results that follows as a straightforward consequence of our abstract theory. At the same time, we extend them to fractional order dispersive equations.

中文翻译：

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带点涡的孤立水波的稳定性

本文研究了具有淹没点涡的自由边界欧拉方程行波解的稳定性。我们证明了具有足够小的涡流强度的足够小幅度的波是条件轨道稳定的。在获得这个结果的过程中，我们为一大类无限维哈密顿系统在对称性存在下的束缚态解发展了一个相当普遍的稳定性/不稳定性理论。这符合 Grillakis、Shatah 和 Strauss 开创性工作的精神，但假设在点涡系统和更广泛的其他流体动力学应用所需的多种方式中得到放松。特别是，我们能够允许泊松图仅具有密集范围，而不是满射，并且是状态相关的。作为一般理论的第二个应用，我们考虑一系列非线性色散偏微分方程，其中包括广义 KdV 和 Benjamin--Ono 方程。这些系统的孤立波的稳定性/不稳定性已被广泛研究，特别是 Bona、Souganidis 和 Strauss，他们使用了 GSS 方法的修改。我们为这些结果提供了一个新的、更直接的证明，这是我们抽象理论的直接结果。同时，我们将它们扩展到分数阶色散方程。我们为这些结果提供了一个新的、更直接的证明，这是我们抽象理论的直接结果。同时，我们将它们扩展到分数阶色散方程。我们为这些结果提供了一个新的、更直接的证明，这是我们抽象理论的直接结果。同时，我们将它们扩展到分数阶色散方程。