Corpus ID: 119692201

# Convexity of Whitham's highest cusped wave

@article{Enciso2018ConvexityOW,
title={Convexity of Whitham's highest cusped wave},
author={A. Enciso and Javier G'omez-Serrano and B. Vergara},
journal={arXiv: Analysis of PDEs},
year={2018}
}
• Published 2018
• Mathematics
• arXiv: Analysis of PDEs
We prove the existence of a periodic traveling wave of extreme form of the Whitham equation that has a convex profile between consecutive stagnation points, at which it is known to feature a cusp of exactly $C^{1/2}$ regularity. The convexity of Whitham's highest cusped wave had been conjectured by Ehrnstr\"om and Wahl\'en.
3 Citations
Waves of maximal height for a class of nonlocal equations with inhomogeneous symbols
In this paper, we consider a class of nonlocal equations where the convolution kernel is given by a Bessel potential symbol of order $\alpha$ for $\alpha > 1$. Based on the properties of theExpand
Global bifurcation of solitary waves for the Whitham equation
• Mathematics, Physics
• 2020
The Whitham equation is a nonlocal shallow water-wave model which combines the quadratic nonlinearity of the KdV equation with the linear dispersion of the full water wave problem. WhithamExpand
Solitary waves in dispersive evolution equations of Whitham type with nonlinearities of mild regularity
We show existence of small solitary and periodic traveling-wave solutions in Sobolev spaces ${\mathrm{H}^s}$, ${ s > 0 }$, to a class of nonlinear, dispersive evolution equations of the formExpand

#### References

SHOWING 1-10 OF 22 REFERENCES
Convexity of Stokes Waves of Extreme Form
• Mathematics
• 2004
Existence is established of a piecewise-convex, periodic, planar curve S below which is defined a harmonic function which simultaneously satisfies prescribed Dirichlet and Neumann boundary conditionsExpand
On the existence of a wave of greatest height and Stokes’s conjecture
• J. Toland
• Mathematics
• Proceedings of the Royal Society of London. A. Mathematical and Physical Sciences
• 1978
It is shown that there exists a solution of Nekrasov’s integral equation which corresponds to the existence of a wave of greatest height and of permanent form moving on the surface of anExpand
Global Bifurcation for the Whitham Equation
• Mathematics, Physics
• 2013
We prove the existence of a global bifurcation branch of 2π -periodic, smooth, traveling-wave solutions of the Whitham equation. It is shown that any subset of solutions in the global branch containsExpand
Wave breaking in the Whitham equation
Abstract We prove wave breaking — bounded solutions with unbounded derivatives — in the nonlinear nonlocal equation which combines the dispersion relation of water waves and a nonlinearity of theExpand
On the existence and stability of solitary-wave solutions to a class of evolution equations of Whitham type
• Mathematics, Physics
• 2012
We consider a class of pseudodifferential evolution equations of the form in which L is a linear smoothing operator and n is at least quadratic near the origin; this class includes in particular theExpand
Existence of a Highest Wave in a Fully Dispersive Two-Way Shallow Water Model
• Mathematics, Physics
• 2016
We consider the existence of periodic traveling waves in a bidirectional Whitham equation, combining the full two-way dispersion relation from the incompressible Euler equations with a canonicalExpand
Traveling waves for the Whitham equation
• Mathematics
• 2009
The existence of traveling waves for the original Whitham equation is investigated. This equation combines a generic nonlinear quadratic term with the exact linear dispersion relation of surfaceExpand
Rigorous Numerics for ill-posed PDEs: Periodic Orbits in the Boussinesq Equation
• Mathematics
• 2015
In this paper, we develop computer-assisted techniques for the analysis of periodic orbits of ill-posed partial differential equations. As a case study, our proposed method is applied to theExpand
Global smooth solutions for the inviscid SQG equation
• Mathematics
• 2016
In this paper, we show the existence of the first non trivial family of classical global solutions of the inviscid surface quasi-geostrophic equation.
An A Posteriori KAM Theorem for Whiskered Tori in Hamiltonian Partial Differential Equations with Applications to some Ill-Posed Equations
• Mathematics
• 2016
The goal of this paper is to develop a KAM theory for tori with hyperbolic directions, which applies to Hamiltonian partial differential equations, even to some ill-posed ones. The main result has anExpand