Nonlinear orbital stability of small solitary gravity waves
Consider two-dimensional irrotational flow of water of unit rest depth under unit gravity, with no surface tension, in the Zakharov–Craig–Sulem form: the surface elevation $\eta$ and surface velocity potential $\psi$ on $\mathbb{R}$ satisfy $$\eta_t = G(\eta)\psi, \qquad \psi_t = -\eta - \frac{1}{2}\psi_x^2 + \frac{(G(\eta)\psi + \eta_x \psi_x)^2}{2(1 + \eta_x^2)},$$ where $G(\eta)\psi = (\phi_y - \eta_x \phi_x)|_{y = \eta}$ is the Dirichlet–Neumann operator, $\Delta \phi = 0$ in $-1 < y < \eta(x)$, $\phi_y = 0$ at $y = -1$ and $\phi = \psi$ at $y = \eta$. Potentials are taken modulo constants. Let $Q_c = (\eta_c, \psi_c)$, $c > 1$, be the classical branch of smooth solitary waves of elevation bifurcating from zero at $c = 1$, so that $Q_c(x - ct)$ is a solution and $(\eta_c, \psi_c')$ decays at infinity. Let $K$ be the Fourier multiplier $\widehat{Kf}(\xi) = \sqrt{|\xi| \tanh|\xi|}\, \widehat{f}(\xi)$ and $\|(a, b)\|_{X^s} = \|a\|_{H^s} + \|Kb\|_{H^s}$. The existence of small solitary waves, close to the KdV soliton, was proved by Friedrichs and Hyers and by Beale. Pego and Sun proved that, for small amplitude, the equations linearized about $Q_c$ are asymptotically stable in an exponentially weighted energy norm, away from the two neutral modes of translation and change of speed, and spectrally stable in the unweighted energy norm; Pego and Sun note, after Bona and Sachs, that without surface tension the usual energy–momentum functional is highly indefinite, so that the variational methods behind the conditional orbital stability results with surface tension appear not to apply. Rousset and Sun proved the transverse linear asymptotic stability of these waves in three-dimensional water.
Problem. Does there exist $c_* > 1$ such that, for every $c \in (1, c_*)$ and $\varepsilon > 0$, some $\delta > 0$ has the following property? Every smooth initial state $Q_0$ with positive depth, $\|Q_0 - Q_c\|_{X^s} < \infty$ for all integers $s \ge 0$, and $\|Q_0 - Q_c\|_{X^6} < \delta$ generates a global smooth graph solution $Q(t)$ satisfying $$\sup_{t \ge 0}\, \inf_{a \in \mathbb{R}} \|Q(t) - Q_c(\cdot - a)\|_{X^0} < \varepsilon.$$
Source: Pego and Sun (2016), §1, who raise the nonlinear stability of solitary water waves. Stated by AIM — Open Applied Problems as orbital stability of small solitary gravity waves for smooth data.
Imported from github.com/MColbrook/AIM.
Classification
- Primary
- AP — Analysis of PDEs
- Secondary
- MP
- Keywords
- AIM Open Applied Problems, orbital stability, solitary waves, water waves
References
@article{pego2016,
author = {Pego, Robert L. and Sun, Shu-Ming},
title = {Asymptotic linear stability of solitary water waves},
journal = {Arch. Ration. Mech. Anal.},
volume = {222},
number = {3},
year = {2016},
pages = {1161--1216},
doi = {10.1007/s00205-016-1021-z},
eprint = {1009.0494},
archivePrefix = {arXiv},
}
@article{friedrichs1954,
author = {Friedrichs, K. O. and Hyers, D. H.},
title = {The existence of solitary waves},
journal = {Comm. Pure Appl. Math.},
volume = {7},
number = {3},
year = {1954},
pages = {517--550},
doi = {10.1002/cpa.3160070305},
}
@article{beale1977,
author = {Beale, J. Thomas},
title = {The existence of solitary water waves},
journal = {Comm. Pure Appl. Math.},
volume = {30},
number = {4},
year = {1977},
pages = {373--389},
doi = {10.1002/cpa.3160300402},
}
@article{rousset2025,
author = {Rousset, Fr{\'e}d{\'e}ric and Sun, Changzhen},
title = {Transverse linear stability of one-dimensional solitary gravity water waves},
journal = {Arch. Ration. Mech. Anal.},
volume = {249},
number = {3},
year = {2025},
pages = {36},
doi = {10.1007/s00205-025-02101-3},
eprint = {2402.11115},
archivePrefix = {arXiv},
}
@misc{aim-solitary-gravity-wave-nonlinear-stability,
author = {Brunton, Steve and Colbrook, Matthew J. and de Hoop, Maarten V. and Stepaniants, George and Townsend, Alex and Ward, Rachel},
title = {{Nonlinear orbital stability of small solitary gravity waves}},
howpublished = {AIM --- Open Applied Problems, GitHub repository},
year = {2026},
url = {https://github.com/MColbrook/AIM/blob/24edaadc6bb272737d8da0745c662c992db07b22/problems/231-solitary-gravity-wave-nonlinear-stability.md},
note = {Entry 231 at commit 24edaad, 2026-10-15},
}
Citing this problem
This identifier is permanent. It will not change if the problem is solved, reclassified or withdrawn.
OP-00643, "Nonlinear orbital stability of small solitary gravity waves", clopen.solutions.