Soliton resolution for bounded nonradial energy-critical waves
Let $u$ be a global real strong solution of the focusing energy-critical wave equation $$\partial_t^2 u - \Delta u - u^5 = 0$$ on $\mathbb{R}^{1+3}$, meaning that $(u, \partial_t u) \in C([0,\infty); \dot H^1 \times L^2)$, $u \in L^8([0,T] \times \mathbb{R}^3)$ for every finite $T > 0$, and the equation holds in the Duhamel sense, and assume $$\sup_{t \ge 0} \|(u, \partial_t u)(t)\|_{\dot H^1 \times L^2} < \infty.$$ For a nonzero stationary solution $Q \in \dot H^1(\mathbb{R}^3)$ of $-\Delta Q = Q^5$ and $\ell \in \mathbb{R}^3$ with $|\ell| < 1$, let $Q_\ell$ be its Lorentz boost, $$Q_\ell(s,y) = Q\Bigl(y + (\gamma_\ell - 1)\frac{(y \cdot \ell)\,\ell}{|\ell|^2} - \gamma_\ell\, \ell s\Bigr), \qquad \gamma_\ell = (1 - |\ell|^2)^{-1/2},$$ with $Q_0 = Q$. For radial solutions the stationary solutions that occur are $\pm \lambda^{-1/2} W(x/\lambda)$ with $W(x) = (1 + |x|^2/3)^{-1/2}$, and Duyckaerts, Kenig and Merle proved the decomposition below, with all $\ell_j = 0$, for every radial global solution (2013); for radial solutions bounded in the energy space, soliton resolution was proved in all odd dimensions by Duyckaerts, Kenig and Merle and in all dimensions at least $4$ by Jendrej and Lawrie (2023). Without symmetry, Duyckaerts, Jia, Kenig and Merle proved in dimensions $3$, $4$ and $5$ that every solution bounded in the energy space decouples, along some sequence of times tending to its maximal time of existence, into a finite sum of modulated solitons and a dispersive term (2017). For compactly supported data in dimensions $3$, $4$ and $5$, Collot, Duyckaerts, Kenig and Merle proved that if every soliton in that decomposition is a boosted ground state $\pm W_{\ell}$ and the velocities are pairwise distinct, then the decomposition holds either for all times or up to a collision on infinitely many time intervals, and they excluded the second alternative when there are at most two solitons (2026).
Problem. Must there exist a solution $v$ of the free wave equation $\partial_t^2 v - \Delta v = 0$, an integer $J \ge 0$, and for $1 \le j \le J$ nonzero stationary solutions $Q_j \in \dot H^1$ of $-\Delta Q_j = Q_j^5$, velocities $\ell_j$ with $|\ell_j| < 1$, scales $\lambda_j(t) > 0$ and centres $x_j(t)$ such that $$\Bigl\|(u, \partial_t u)(t) - (v, \partial_t v)(t) - \sum_{j=1}^{J} \Bigl(\lambda_j^{-1/2} Q_{j,\ell_j}\bigl(0, \tfrac{\cdot - x_j}{\lambda_j}\bigr),\ \lambda_j^{-3/2}\, \partial_s Q_{j,\ell_j}\bigl(0, \tfrac{\cdot - x_j}{\lambda_j}\bigr)\Bigr)\Bigr\|_{\dot H^1 \times L^2} \longrightarrow 0$$ as $t \to \infty$ along all times, where $Q_{j,\ell}$ is the Lorentz boost of $Q_j$, with $\lambda_j(t)/t \to 0$, $x_j(t)/t \to \ell_j$ and, for $i \ne j$, $$\frac{\lambda_i}{\lambda_j} + \frac{\lambda_j}{\lambda_i} + \frac{|x_i - x_j|^2}{\lambda_i \lambda_j} \longrightarrow \infty \qquad (t \to \infty)?$$
Source: The soliton resolution conjecture, a long-held belief in mathematical physics (see Duyckaerts, Kenig and Merle (2013), introduction); for nonradial solutions of this equation, see Duyckaerts, Jia, Kenig and Merle (2017) and Collot, Duyckaerts, Kenig and Merle (2026). Stated in this form, for bounded global solutions in three dimensions, by AIM — Open Applied Problems.
Imported from github.com/MColbrook/AIM.
Classification
- Primary
- AP — Analysis of PDEs
- Secondary
- MP
- Keywords
- AIM Open Applied Problems, energy-critical wave equations, soliton resolution
References
@article{duyckaerts2013,
author = {Duyckaerts, Thomas and Kenig, Carlos and Merle, Frank},
title = {Classification of the radial solutions of the focusing, energy-critical wave equation},
journal = {Camb. J. Math.},
volume = {1},
number = {1},
year = {2013},
pages = {75--144},
doi = {10.4310/CJM.2013.v1.n1.a3},
eprint = {1204.0031},
archivePrefix = {arXiv},
}
@article{duyckaerts2017,
author = {Duyckaerts, Thomas and Jia, Hao and Kenig, Carlos and Merle, Frank},
title = {Soliton resolution along a sequence of times for the focusing energy critical wave equation},
journal = {Geom. Funct. Anal.},
volume = {27},
number = {4},
year = {2017},
pages = {798--862},
doi = {10.1007/s00039-017-0418-7},
eprint = {1601.01871},
archivePrefix = {arXiv},
}
@article{collot2026,
author = {Collot, Charles and Duyckaerts, Thomas and Kenig, Carlos and Merle, Frank},
title = {Non-radial soliton resolution for the energy critical wave equation for initial data with compact support},
journal = {Discrete Contin. Dyn. Syst.},
volume = {56},
year = {2026},
pages = {501--556},
doi = {10.3934/dcds.2026115},
}
@article{duyckaerts2023,
author = {Duyckaerts, Thomas and Kenig, Carlos and Merle, Frank},
title = {Soliton resolution for the radial critical wave equation in all odd space dimensions},
journal = {Acta Math.},
volume = {230},
number = {1},
year = {2023},
pages = {1--92},
doi = {10.4310/ACTA.2023.v230.n1.a1},
eprint = {1912.07664},
archivePrefix = {arXiv},
}
@article{jendrej2023,
author = {Jendrej, Jacek and Lawrie, Andrew},
title = {Soliton resolution for the energy-critical nonlinear wave equation in the radial case},
journal = {Ann. PDE},
volume = {9},
number = {2},
year = {2023},
pages = {18},
doi = {10.1007/s40818-023-00159-4},
eprint = {2203.09614},
archivePrefix = {arXiv},
}
@misc{aim-nonradial-soliton-resolution,
author = {Brunton, Steve and Colbrook, Matthew J. and de Hoop, Maarten V. and Stepaniants, George and Townsend, Alex and Ward, Rachel},
title = {{Soliton resolution for bounded nonradial energy-critical waves}},
howpublished = {AIM --- Open Applied Problems, GitHub repository},
year = {2026},
url = {https://github.com/MColbrook/AIM/blob/24edaadc6bb272737d8da0745c662c992db07b22/problems/078-nonradial-soliton-resolution.md},
note = {Entry 078 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-00625, "Soliton resolution for bounded nonradial energy-critical waves", clopen.solutions.