Symmetry for a Dirichlet–Neumann problem arising in water waves

Symmetry for a Dirichlet-neumann Problem Arising in Water Waves

Given a smooth u : R → R, say u = u(y), we consider u = u(x, y) to be a solution of  ∆u = 0 for any (x, y) ∈ (0, 1)× R , u(0, y) = u(y) for any y ∈ R, ux(1, y) = 0 for any y ∈ R. We define the Dirichlet-Neumann operator (Lu)(y) = ux(0, y) and we prove a symmetry result for equations of the form (Lu)(y) = f(u(y)). In particular, bounded, monotone solutions in R are proven to depend only on one...

On Nonlinear Water Waves in a Channel

Symmetry of Solitary Water Waves with Vorticity

Symmetry and monotonicity properties of solitary water-waves of positive elevation with supercritical values of parameter are established for an arbitrary vorticity. The proof uses the detailed knowledge of asymptotic decay of supercritical solitary waves at infinity and the method of moving planes.

Some Criteria for the Symmetry of Stratified Water Waves

This paper considers two-dimensional stably stratified steady periodic gravity water waves with surface profiles monotonic between crests and troughs. We provide sufficient conditions under which such waves are necessarily symmetric. This is done by first exploiting some elliptic structure in the governing equations to show that, in certain size regimes, a maximum principle holds. This then for...

A free boundary problem arising in a model for shallow water entry at small deadrise angles

The impact of a rigid body on liquid is an important free boundary problem with applications ranging in scale from asteroid impact to droplet motion. The severe nonlinearities involved mean that even the basic problem in which the liquid is inviscid and incompressible, the flow starts from rest and is therefore irrotational, and the effects of gravity, surface tension and air cushioning are neg...