On Whitham's conjecture of a highest cusped wave for a nonlocal dispersive equation

We consider the Whitham equation $u_t + 2u u_x+Lu_x = 0$, where L is the nonlocal Fourier multiplier operator given by the symbol $m(\xi) = \sqrt{\tanh \xi /\xi}$. G. B. Whitham conjectured that for this equation there would be a highest, cusped, travelling-wave solution. We find this wave as a limiting case at the end of the main bifurcation curve of $P$-periodic solutions, and give several qualitative properties of it, including its optimal $C^{1/2}$-regularity. An essential part of the proof consists in an analysis of the integral kernel corresponding to the symbol $m(\xi)$, and a following study of the highest wave. In particular, we show that the integral kernel corresponding to the symbol $m(\xi)$ is completely monotone, and provide an explicit representation formula for it.Comment: 40 pages, 3 figures. This version is identical to the one accepted for publication in Annales de l'Institut Henri Poincare, Analyse non lineair

Trimodal steady water waves

We construct three-dimensional families of small-amplitude gravity-driven rotational steady water waves on finite depth. The solutions contain counter-currents and multiple crests in each minimal period. Each such wave generically is a combination of three different Fourier modes, giving rise to a rich and complex variety of wave patterns. The bifurcation argument is based on a blow-up technique, taking advantage of three parameters associated with the vorticity distribution, the strength of the background stream, and the period of the wave.Comment: The final publication is available at Springer via http://dx.doi.org/10.1007/s00205-014-0812-

Steady three-dimensional rotational flows: an approach via two stream functions and Nash-Moser iteration

We consider the stationary flow of an inviscid and incompressible fluid of constant density in the region $D=(0, L)\times \mathbb{R}^2$. We are concerned with flows that are periodic in the second and third variables and that have prescribed flux through each point of the boundary $\partial D$. The Bernoulli equation states that the "Bernoulli function" $H:= \frac 1 2 |v|^2+p$ (where $v$ is the velocity field and $p$ the pressure) is constant along stream lines, that is, each particle is associated with a particular value of $H$. We also prescribe the value of $H$ on $\partial D$. The aim of this work is to develop an existence theory near a given constant solution. It relies on writing the velocity field in the form $v=\nabla f\times \nabla g$ and deriving a degenerate nonlinear elliptic system for $f$ and $g$. This system is solved using the Nash-Moser method, as developed for the problem of isometric embeddings of Riemannian manifolds; see e.g. the book by Q. Han and J.-X. Hong (2006). Since we can allow $H$ to be non-constant on $\partial D$, our theory includes three-dimensional flows with non-vanishing vorticity

Steady water waves with multiple critical layers

We construct small-amplitude periodic water waves with multiple critical layers. In addition to waves with arbitrarily many critical layers and a single crest in each period, two-dimensional sets of waves with several crests and troughs in each period are found. The setting is that of steady two-dimensional finite-depth gravity water waves with vorticity.Comment: 16 pages, 2 figures. As accepted for publication in SIAM J. Math. Ana

Periodic solitons for the elliptic-elliptic focussing Davey-Stewartson equations

We consider the elliptic-elliptic, focussing Davey-Stewartson equations, which have an explicit bright line soliton solution. The existence of a family of periodic solitons, which have the profile of the line soliton in the longitudinal spatial direction and are periodic in the transverse spatial direction, is established using dynamical systems arguments. We also show that the line soliton is linearly unstable with respect to perturbations in the transverse direction.Comment: arXiv admin note: text overlap with arXiv:1411.247

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

We consider two-dimensional steady periodic gravity waves on water of finite depth with a prescribed but arbitrary vorticity distribution. The water surface is allowed to be overhanging and no assumptions regarding the absence of stagnation points and critical layers are made. Using conformal mappings and a new Babenko-type reformulation of Bernoulli's equation, we uncover an equivalent formulation as "identity plus compact", which is amenable to Rabinowitz' global bifurcation theorem. This allows us to construct a global connected set of solutions, bifurcating from laminar flows with a flat surface. Moreover, a nodal analysis is carried out for these solutions under a monotonicity assumption on the vorticity function.Comment: arXiv admin note: substantial text overlap with arXiv:2109.0607

Generalized modulation theory for strongly nonlinear gravity waves in a compressible atmosphere

This study investigates nonlinear gravity waves in the compressible atmosphere from the Earth's surface to the deep atmosphere. These waves are effectively described by Grimshaw's dissipative modulation equations which provide the basis for finding stationary solutions such as mountain lee waves and testing their stability in an analytic fashion. Assuming energetically consistent boundary and far-field conditions, that is no energy flux through the surface, free-slip boundary, and finite total energy, general wave solutions are derived and illustrated in terms of realistic background fields. These assumptions also imply that the wave-Reynolds number must become less than unity above a certain height. The modulational stability of admissible, both non-hydrostatic and hydrostatic, waves is examined. It turns out that, when accounting for the self-induced mean flow, the wave-Froude number has a resonance condition. If it becomes $1/\sqrt{2}$, then the wave destabilizes due to perturbations from the essential spectrum of the linearized modulation equations. However, if the horizontal wavelength is large enough, waves overturn before they can reach the modulational stability condition