Global smooth propagation for a laser envelope coupled to ionization
Write $x = (x_1, x_2, z) \in \mathbb{R}^3$. Fix $\varepsilon, c, c_g, \alpha_4 > 0$, $\alpha_5 \ge 0$ and an integer $K \ge 2$, and consider a complex envelope $u$ and a real electron density $\rho$ satisfying $$ \begin{aligned} i(\partial_t + c_g \partial_z) u + \varepsilon \Delta u + \varepsilon (|u|^2 - \rho) u &= -i \varepsilon c \bigl(\alpha_4 |u|^{2K-2} u + \alpha_5 \rho u\bigr), \\ \partial_t \rho &= \varepsilon \alpha_4 |u|^{2K} + \varepsilon \alpha_5 \rho |u|^2 . \end{aligned} $$ The Laplacian is elliptic in all three spatial coordinates, which corresponds to anomalous group-velocity dispersion.
Without ionization ($\alpha_4 = \alpha_5 = 0$, $\rho = 0$ initially) the system reduces, in a frame moving at speed $c_g$ along $z$ and in the time $\varepsilon t$, to the focusing cubic nonlinear Schrödinger equation, which in three dimensions has solutions that blow up in finite time. Dumas, Lannes and Szeftel proved local existence in $H^N \times H^N$ for every integer $N > 3/2$, and observed that the $L^2$ norm of $u$ is dissipated while the energy is neither conserved nor dissipated.
Problem. For every Schwartz initial pair $(u_0, \rho_0)$ with $\rho_0 \ge 0$, must the local solution extend to all $t \ge 0$, with $(u, \rho) \in C([0, \infty); H^m(\mathbb{R}^3))$ for every integer $m \ge 3$?
Source: Dumas, Lannes and Szeftel (2016), Open Problem 3, for their systems (54) and (55) in dimensions 2 and 3. Stated by AIM — Open Applied Problems for system (54) in three dimensions with anomalous dispersion and Schwartz data.
Imported from github.com/MColbrook/AIM.
Classification
- Primary
- AP — Analysis of PDEs
- Secondary
- MP
- Keywords
- AIM Open Applied Problems, laser filamentation, nonlinear Schrödinger equations
References
@incollection{dumas2016variants,
author = {Dumas, {\'E}ric and Lannes, David and Szeftel, J{\'e}r{\'e}mie},
title = {Variants of the focusing {NLS} equation: derivation, justification, and open problems related to filamentation},
booktitle = {Laser Filamentation: Mathematical Methods and Models},
editor = {Bandrauk, Andr{\'e} D. and Lorin, Emmanuel and Moloney, Jerome V.},
series = {CRM Series in Mathematical Physics},
publisher = {Springer},
address = {Cham},
year = {2016},
pages = {19--75},
doi = {10.1007/978-3-319-23084-9_2},
}
@article{couairon2007femtosecond,
author = {Couairon, A. and Mysyrowicz, A.},
title = {Femtosecond filamentation in transparent media},
journal = {Phys. Rep.},
volume = {441},
number = {2--4},
year = {2007},
pages = {47--189},
doi = {10.1016/j.physrep.2006.12.005},
}
@misc{aim-ionization-laser-envelope-global-regularity,
author = {Brunton, Steve and Colbrook, Matthew J. and de Hoop, Maarten V. and Stepaniants, George and Townsend, Alex and Ward, Rachel},
title = {{Global smooth propagation for a laser envelope coupled to ionization}},
howpublished = {AIM --- Open Applied Problems, GitHub repository},
year = {2026},
url = {https://github.com/MColbrook/AIM/blob/24edaadc6bb272737d8da0745c662c992db07b22/problems/441-ionization-laser-envelope-global-regularity.md},
note = {Entry 441 at commit 24edaad, 2026-10-15},
}
Resolution claims
We construct a finite-time singularity for a three-dimensional focusing Schrödinger envelope coupled to an ionization density. For one strictly positive ionization coefficient, with cubic absorption and no avalanche term, a nonzero compactly supported smooth optical datum and zero initial density produce a smooth solution whose gradient norm tends to infinity at a finite time. The density remains nonnegative. Thus global …
Status history
| When | Change | By | Reason |
|---|---|---|---|
| 2026-10-18 | Needs verification → Solution claimed | — | Recorded here: AIM @ 24edaad, paper/ionization_blowup.pdf. |
Citing this problem
This identifier is permanent. It will not change if the problem is solved, reclassified or withdrawn.
OP-00803, "Global smooth propagation for a laser envelope coupled to ionization", clopen.solutions.