Global L2 scattering for the planar hyperbolic cubic Schrödinger equation
For $u_0 \in L^2(\mathbb{R}^2)$ consider the cubic hyperbolic Schrödinger equation $$i\partial_t u + \partial_x^2 u - \partial_y^2 u + |u|^2 u = 0, \qquad u(0) = u_0,$$ on $\mathbb{R} \times \mathbb{R}^2$: the scalar equation, without the nonlocal term of the Davey–Stewartson systems. The linear flow $e^{it(\partial_x^2 - \partial_y^2)}$ satisfies the same Strichartz estimates as the standard Schrödinger flow, in particular $\|e^{it(\partial_x^2 - \partial_y^2)} f\|_{L^4(\mathbb{R} \times \mathbb{R}^2)} \le C \|f\|_{L^2}$, so by standard arguments the problem, which is mass-critical, is locally well-posed for all data in $L^2$ and has global solutions for small data. Dodson, Marzuola, Pausader and Spirn proved a profile decomposition in $L^2$ for this linear flow.
Problem. Does the local solution, unique in $C(I; L^2) \cap L^4(I \times \mathbb{R}^2)$ on compact time intervals $I$, extend for every $u_0 \in L^2(\mathbb{R}^2)$ to a global solution $u \in C(\mathbb{R}; L^2) \cap L^4_{\mathrm{loc}}(\mathbb{R} \times \mathbb{R}^2)$ that depends continuously on $u_0$ on compact time intervals, and are there $u_\pm \in L^2(\mathbb{R}^2)$ such that $$\lim_{t \to \pm\infty} \bigl\| u(t) - e^{it(\partial_x^2 - \partial_y^2)} u_\pm \bigr\|_{L^2} = 0 \, ?$$
Source: Saut and Wang (2024), Conjecture 1; stated earlier, without a source, as global well-posedness and scattering for all L² data, by Dodson, Marzuola, Pausader and Spirn (2018).
Imported from github.com/MColbrook/AIM.
Classification
- Primary
- AP — Analysis of PDEs
- Secondary
- MP
- Keywords
- AIM Open Applied Problems, dispersive equations, nonlinear Schrödinger equations, scattering
References
@article{saut2024,
author = {Saut, Jean-Claude and Wang, Yuexun},
title = {On the hyperbolic nonlinear {S}chr{\"o}dinger equations},
journal = {Adv. Contin. Discrete Models},
volume = {2024},
year = {2024},
pages = {15},
doi = {10.1186/s13662-024-03811-w},
}
@article{dodson2018,
author = {Dodson, Benjamin and Marzuola, Jeremy L. and Pausader, Benoit and Spirn, Daniel P.},
title = {The profile decomposition for the hyperbolic {S}chr{\"o}dinger equation},
journal = {Illinois J. Math.},
volume = {62},
number = {1-4},
year = {2018},
pages = {293--320},
doi = {10.1215/ijm/1552442664},
eprint = {1708.08014},
archivePrefix = {arXiv},
note = {Erratum: Illinois J. Math. 65 (2021), no. 1, 259--260, doi:10.1215/00192082-8886975},
}
@misc{aim-hyperbolic-cubic-nls-scattering,
author = {Brunton, Steve and Colbrook, Matthew J. and de Hoop, Maarten V. and Stepaniants, George and Townsend, Alex and Ward, Rachel},
title = {{Global L2 scattering for the planar hyperbolic cubic Schrödinger equation}},
howpublished = {AIM --- Open Applied Problems, GitHub repository},
year = {2026},
url = {https://github.com/MColbrook/AIM/blob/a153f469ee0ea64e876b2ad97c79d252c19a3e22/problems/421-hyperbolic-cubic-nls-scattering.md},
note = {Entry 421 at commit a153f46, 2026-10-13},
}
Citing this problem
This identifier is permanent. It will not change if the problem is solved, reclassified or withdrawn.
OP-00442, "Global L2 scattering for the planar hyperbolic cubic Schrödinger equation", clopen.solutions.