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Abstract This paper considers two-dimensional steady solitary waves with constant vorticity propagating under the influence of gravity over an impermeable flat bed. Unlike in previous works on solitary waves, we allow for both internal stagnation points and overhanging wave profiles. Using analytic global bifurcation theory, we construct continuous curves of large-amplitude solutions. Along these curves, either the wave amplitude approaches the maximum possible value, the dimensionless wave speed becomes unbounded, or a singularity develops in a conformal map describing the fluid domain. This is stronger than what one would expect from a straightforward generalization of existing results for periodic waves. We also show that an arbitrary solitary wave of elevation with constant vorticity must be supercritical. The existence proof relies on a novel reformulation of the problem as an elliptic system for two scalar functions in a fixed domain, one describing the conformal map of the fluid region and the other the flow beneath the wave. 

Abstract In this paper, we present a center manifold reduction theorem for quasilinear elliptic equations posed on infinite cylinders that is done without a phase space in the sense that we avoid explicitly reformulating the PDE as an evolution problem. Under suitable hypotheses, the resulting center manifold is finite dimensional and captures all sufficiently small bounded solutions. Compared with classical methods, the reduced ODE on the manifold is more directly related to the original physical problem and also easier to compute. The analysis is conducted directly in Hölder spaces, which is often desirable for elliptic equations. We then use this machinery to construct small bounded solutions to a variety of systems. These include heteroclinic and homoclinic solutions of the anti-plane shear problem from nonlinear elasticity; exact slow moving invasion fronts in a two-dimensional Fisher–KPP equation; and hydrodynamic bores with vorticity in a channel. The last example is particularly interesting in that we find solutions with critical layers and distinctive ‘half cat’s eye’ streamline patterns.