Finite-time collapse in the planar self-steepening Schrödinger equation
On $\mathbb{R}^2$ consider the focusing cubic Schrödinger equation with a self-steepening term in the $y$-direction, $$i\partial_t u + \partial_x^2 u + \partial_y^2 u + |u|^2 u + i\partial_y(|u|^2 u) = 0, \qquad u(0) = u_0.$$ In their study of variants of the focusing cubic NLS equation for laser beams, Dumas, Lannes and Szeftel considered $i\partial_t v + \Delta v + (1 + i\varepsilon\alpha \cdot \nabla)(|v|^2 v) = 0$, whose derivative term may account for the self-steepening of the pulse, and asked for a description of the blow-up scenarios of its solutions that blow up in finite time (2016); the equation above is the case $\varepsilon\alpha = (0, 1)$. In numerical simulations of this equation, Arbunich, Klein and Sparber found apparent finite-time blow-up for the Gaussian datum $u_0 = 4e^{-(x^2 + y^2)}$, and no indication of blow-up up to time $5$ for a perturbed stationary state (2019).
Problem. Do there exist $u_0 \in \mathcal{S}(\mathbb{R}^2; \mathbb{C})$ and $0 < T < \infty$ such that the equation has a classical solution on $[0, T)$, belonging to $C([0, T); H^m(\mathbb{R}^2))$ for every integer $m \ge 3$, with $$\limsup_{t \uparrow T} \|u(t)\|_{L^\infty(\mathbb{R}^2)} = \infty?$$ The coefficient of the derivative term is fixed and nonzero: the question asks for a singular solution of this two-dimensional equation, not of a dispersionless approximation or a periodic numerical discretization.
Source: Stated by AIM — Open Applied Problems, after Dumas, Lannes and Szeftel (2016), Open Problem 5 (blow-up scenarios for the self-steepening cubic NLS), and the numerical simulations of Arbunich, Klein and Sparber (2019), §3.4.
Imported from github.com/MColbrook/AIM.
Classification
- Primary
- AP — Analysis of PDEs
- Secondary
- MP
- Keywords
- AIM Open Applied Problems, derivative nonlinear Schrödinger equations, finite-time blow-up
References
@incollection{dumas2016,
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},
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},
eprint = {1405.7308},
archivePrefix = {arXiv},
}
@article{arbunich2019,
author = {Arbunich, Jack and Klein, Christian and Sparber, Christof},
title = {On a class of derivative nonlinear {S}chr{\"o}dinger-type equations in two spatial dimensions},
journal = {ESAIM Math. Model. Numer. Anal.},
volume = {53},
number = {5},
year = {2019},
pages = {1477--1505},
doi = {10.1051/m2an/2019018},
eprint = {1805.12351},
archivePrefix = {arXiv},
}
@misc{aim-planar-self-steepening-finite-time-collapse,
author = {Brunton, Steve and Colbrook, Matthew J. and de Hoop, Maarten V. and Stepaniants, George and Townsend, Alex and Ward, Rachel},
title = {{Finite-time collapse in the planar self-steepening Schrödinger equation}},
howpublished = {AIM --- Open Applied Problems, GitHub repository},
year = {2026},
url = {https://github.com/MColbrook/AIM/blob/24edaadc6bb272737d8da0745c662c992db07b22/problems/442-planar-self-steepening-finite-time-collapse.md},
note = {Entry 442 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-00633, "Finite-time collapse in the planar self-steepening Schrödinger equation", clopen.solutions.