An extreme solitary wave for the bidirectional Whitham system
Let $K$ be the convolution kernel on $\mathbb{R}$ with Fourier transform $\widehat K(\xi) = \tanh(\xi)/\xi$, extended continuously by $\widehat K(0) = 1$, and consider the bidirectional Whitham (Whitham–Boussinesq) system $$\partial_t \eta + \partial_x (K * v + \eta v) = 0, \qquad \partial_t v + \partial_x\bigl(\eta + \tfrac12 v^2\bigr) = 0.$$ A profile $v$ solving $$K * v = c^2 v - \tfrac{3c}{2} v^2 + \tfrac12 v^3$$ gives, with $\eta = cv - v^2/2$, a traveling solution $(\eta(x - ct), v(x - ct))$; the crest height $v(0) = c(1 - 1/\sqrt{3})$ corresponds to $\eta(0) = c^2/3$. Ehrnström, Johnson and Claassen obtained a highest, cusped, $2\pi$-periodic traveling wave of this system as a limiting case at the end of the main bifurcation branch of periodic traveling waves. Nilsson and Wang constructed small solitary waves for a class of Whitham–Boussinesq systems that includes this one, and Ehrnström, Mæhlen and Varholm determined the exact leading-order behaviour at the crest of the cusped highest waves.
Problem. Do there exist $c > 1$ and an even continuous function $v : \mathbb{R} \to (0, \infty)$, smooth on $\mathbb{R} \setminus \{0\}$, strictly decreasing on $(0, \infty)$ and tending to zero at infinity, such that $$K * v = c^2 v - \tfrac{3c}{2} v^2 + \tfrac12 v^3 \quad \text{on } \mathbb{R}, \qquad v(0) = c\left(1 - \frac{1}{\sqrt{3}}\right)?$$
Source: Stated by AIM — Open Applied Problems, after Ehrnström, Mæhlen and Varholm (2025), Remark 4.8, which notes that no existence result for extreme solitary solutions of this system is available.
Imported from github.com/MColbrook/AIM.
Classification
- Primary
- AP — Analysis of PDEs
- Secondary
- MP
- Keywords
- AIM Open Applied Problems, highest waves, solitary waves, Whitham–Boussinesq system
References
@article{ehrnstrom2025,
author = {Ehrnstr{\"o}m, Mats and M{\ae}hlen, Ola I. H. and Varholm, Kristoffer},
title = {On the precise cusped behaviour of extreme solutions to {W}hitham-type equations},
journal = {Ann. Inst. H. Poincar{\'e} C Anal. Non Lin{\'e}aire},
volume = {42},
number = {1},
year = {2025},
pages = {125--162},
doi = {10.4171/AIHPC/104},
eprint = {2302.08856},
archivePrefix = {arXiv},
}
@article{ehrnstrom2019,
author = {Ehrnstr{\"o}m, Mats and Johnson, Mathew A. and Claassen, Kyle M.},
title = {Existence of a highest wave in a fully dispersive two-way shallow water model},
journal = {Arch. Ration. Mech. Anal.},
volume = {231},
number = {3},
year = {2019},
pages = {1635--1673},
doi = {10.1007/s00205-018-1306-5},
eprint = {1610.02603},
archivePrefix = {arXiv},
}
@article{nilsson2019,
author = {Nilsson, Dag and Wang, Yuexun},
title = {Solitary wave solutions to a class of {W}hitham--{B}oussinesq systems},
journal = {Z. Angew. Math. Phys.},
volume = {70},
number = {3},
year = {2019},
pages = {70},
doi = {10.1007/s00033-019-1116-0},
eprint = {1810.03405},
archivePrefix = {arXiv},
}
@misc{aim-extreme-bidirectional-solitary-wave,
author = {Brunton, Steve and Colbrook, Matthew J. and de Hoop, Maarten V. and Stepaniants, George and Townsend, Alex and Ward, Rachel},
title = {{An extreme solitary wave for the bidirectional Whitham system}},
howpublished = {AIM --- Open Applied Problems, GitHub repository},
year = {2026},
url = {https://github.com/MColbrook/AIM/blob/24edaadc6bb272737d8da0745c662c992db07b22/problems/162-extreme-bidirectional-solitary-wave.md},
note = {Entry 162 at commit 24edaad, 2026-10-15},
}
Resolution claims
We construct an extreme solitary wave for the bidirectional Whitham system. Its velocity profile is positive, even, continuous, smooth away from its crest and strictly decreasing to zero on the positive half-line. The crest reaches the first critical value of the cubic traveling-wave equation. The construction starts with positive highest periodic waves of arbitrarily large period. A common modulus of …
Status history
| When | Change | By | Reason |
|---|---|---|---|
| 2026-10-18 | Needs verification → Solution claimed | — | Recorded here: AIM @ 24edaad, paper/manuscript.pdf. |
Citing this problem
This identifier is permanent. It will not change if the problem is solved, reclassified or withdrawn.
OP-00789, "An extreme solitary wave for the bidirectional Whitham system", clopen.solutions.