On the Stability of Solitary Water Waves with a Point Vortex

Abstract

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 (GSS) [20], 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 Korteweg-de Vries (KdV) and Benjamin-Ono equations. The stability or instability of solitary waves for these systems has been studied extensively, notably by Bona, Souganidis, and Strauss [6], who used a modification of the GSS method. We provide a new, more direct proof of these results, as a straightforward consequence of our abstract theory. At the same time, we allow fractional dispersion, and obtain a new instability result for fractional KdV.

On the Existence and Instability of Solitary Water Waves with a Finite Dipole

“…In this paper, we will use a recent variant of the GSS method developed by Varholm, Wahlén, and Walsh [30]. Among other improvements, this machinery permits J to have merely a dense range, and also allow for a mismatch between the space where the problem is well-posed and the natural energy space.…”

“…For the convenience of the reader, several appendices have been included. Appendix A contains a summary of the instability theory developed by Varholm-Wahlén-Walsh [30]. We also derive steady and unsteady equations for the velocity potential and stream functions in Appendix B.…”

“…In this section, we show that the traveling waves constructed in Theorem 1.1 are orbitally unstable. To do so, we follow the general strategy of Varholm-Wahlén-Walsh [30] which is an adaptation of the classical Grillakis-Shatah-Strauss method [10,11]. In subsection 3.1, we rewrite the equations of capillary-gravity waves with a finite dipole (1.1)-(1.4) as a Hamiltonian system and give an explicit form for its energy and momentum.…”

“…The next lemma records the fact that the momentum and the energy are conserved. with The proof of this lemma is done by nearly identical arguments to that of [30,Lemma 5.4], as the symmetry groups are essentially the same. We therefore omit the details.…”

“…This section summarizes the instability theory developed by Varholm, Wahlén, and Walsh in [30,Sections 2,4]. We are considering the stability property of an abstract Hamiltonian…”