  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.
