Abstract. We establish the existence of small-amplitude uni-and bimodal steady periodic gravity waves with an affine vorticity distribution, using a bifurcation argument that differs slightly from earlier theory. The solutions describe waves with critical layers and an arbitrary number of crests and troughs in each minimal period. An important part of the analysis is a fairly complete description of the local geometry of the so-called kernel equation, and of the small-amplitude solutions. Finally, we investigate the asymptotic behavior of the bifurcating solutions.

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.

Analytic global bifurcation theory is used to construct a large variety of families of steady periodic two-dimensional gravity water waves with real-analytic vorticity distributions, propagating in an incompressible fluid. The waves that are constructed can possess an arbitrary number of interior stagnation points in the fluid and corresponding critical layers consisting of closed streamlines. This is made possible by the use of the so-called naive flattening transform, which has previously only been used for local bifurcation.

We construct small-amplitude solitary traveling gravity-capillary water waves with a finite number of point vortices along a vertical line, on finite depth. This is done using a local bifurcation argument. The properties of the resulting waves are also examined: We find that they depend significantly on the position of the point vortices in the water column.Comment: 36 pages, 4 figures. As accepted for publication in Discrete Cont. Dyn. Sys

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Analytic global bifurcation theory is used to construct a large variety of families of steady periodic two-dimensional gravity water waves with real-analytic vorticity distributions, propagating in an incompressible fluid. The waves that are constructed can possess an arbitrary number of interior stagnation points in the fluid, and corresponding critical layers consisting of closed streamlines. This is made possible by the use of the so-called naive flattening transform, which has previously only been used for local bifurcation.