Global bifurcation and highest waves on water of finite depth

Abstract

We consider the two-dimensional problem for steady water waves on finite depth with vorticity. While neglecting the effects of surface tension we construct connected families of large amplitude periodic waves approaching either a solitary wave, the highest solitary wave or the highest Stokes wave. In contrast to previous studies we fix the Bernoulli constant and consider the wavelength as a bifurcation parameter, which guarantees that the limiting wave has a finite depth. In fact, this is the first rigorous proof of the existence of extreme Stokes waves on water of finite depth. Beside the existence of highest waves we provide a new result about the regularity of Stokes waves of arbitrary amplitude (including extreme waves).

The Subharmonic Bifurcation of Stokes Waves on Vorticity Flow

“…Then either (i) C ′ δ is unbounded in R × X, or (ii) C ′ δ contains another trivial point (t 1 , 0) with t 1 = t M , or (iii) C ′ δ contains a point (t, w) ∈ ∂O δ . In [15] it is proved an estimate Λ(t) ≥ c 0 > 0 for all t ∈ R, where the constant c 0 depends only on ω and r. This estimate implies λ(t) ≤ Λ(0) c 0 .…”

On Loops in Water Wave Branches and Monotonicity of Water Waves

“…The only difference here is that the bifurcation parameter is the period and in verification of compactness and Fredholm properties in the global bifurcation theorem must be changed a little bit. We refer also to [9] and [10] where the assertion is proved for the unidirectional flows.…”

Large-amplitude steady gravity water waves with general vorticity and critical layers

“…In order to overcome the inherited downsides of the above-mentioned semi-hodograph transformation and to allow for stagnation points and critical layers, many papers use a "naive" flattening transform, where on each vertical ray the vertical coordinate is scaled to a constant; see [1,21,23,24,31,35,37,47,50]. Consequently, a drawback of this naive flattening is that it needs the surface profile to be a graph and can thus not allow for overhanging waves.…”