Traveling water waves — the ebb and flow of two centuries

Abstract

This survey covers the mathematical theory of steady water waves with an emphasis on topics that are at the forefront of current research. These areas include: variational characterizations of traveling water waves; analytical and numerical studies of periodic waves with critical layers that may overhang; existence, nonexistence, and qualitative theory of solitary waves and fronts; traveling waves with localized vorticity or density stratification; and waves in three dimensions.

Highest Cusped Waves for the Burgers-Hilbert equation

“…In all the above cases we could reduce the problem of enclosing the Clausen functions on x to evaluating them on, a subset of, x = 0, x = x (or −x if x < 0), x = x, x = x 0 (the midpoint) and x = π. In all cases except x = 0 we have 0 < x < 2π so we can use (23) and (24).…”

“…The study of traveling waves is an important topic in fluid dynamics, see e.g. [23] for a recent overview of traveling water waves. The traveling wave assumption f (x, t) = ϕ(x − ct), where c > 0 denotes the wave speed, gives us…”

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

“…In the previous century, mostly irrotational flows were considered, which enjoy the advantage of being thoroughly treatable by tools of complex analysis, because the water velocity can be written as the gradient of a harmonic potential. We refer to [27,28,45] for surveys on irrotational water waves. However, in many situations it is important to take vorticity into account, for example, in the presence of underlying non-uniform currents.…”

“…Then, when one wants to allow for stagnation points and critical layers, and also for overhanging waves, the typical strategy is to use a conformal change of variables. This is classical in the irrotational context and can be used to reformulate the problem either as a singular integral equation for the tangent angle of the free surface [27,28,45] or as a nonlinear pseudodifferential equation for the surface elevation in the new variables introduced by Babenko [5] (see also [27,28] and reference therein). The latter approach was extended to constant vorticity by Constantin and Vărvărucă in [14] and then utilised, together with Strauss, in [13] to obtain pure gravity water waves of large amplitude.…”

Traveling wave solutions to the inclined or periodic free boundary incompressible Navier-Stokes equations

“…For a brief survey of the subject, we refer to Section 1.2 of Leoni-Tice [18] and Section 1.4 of Tice [28]. For a more thorough review of the subject, we refer the surveys of Tolland [29], Groves [10], Strauss [26], the recent paper by Strauss et al [15] in the inviscid case, and to the surveys of Zadrzyńska [32] and Shibata-Shimizu [23] in the viscous case. When κ = 0, the small data theory for the free boundary problem (1.3) over periodic domains is well-understood in dimension n = 3.…”