Intercritical norm inflation for the dispersion-managed Schrödinger equation
On $\mathbb{R}^3$ consider the defocusing quintic Gabitov–Turitsyn equation $$i \partial_t u - \Delta u + \int_0^1 e^{-i\sigma\Delta}\bigl(|e^{i\sigma\Delta} u|^4 e^{i\sigma\Delta} u\bigr)\,d\sigma = 0,$$ where $e^{i\sigma\Delta}$ is the free Schrödinger group, and let $\|f\|_{\dot H^{1/4}}^2 = \int |\xi|^{1/2} |\widehat f(\xi)|^2\,d\xi$. This averaged equation models pulses in dispersion-managed optical fibres. Kowalski identified two critical regularities for it, here $s_m = 1$ and the integrated-scaling index $s_i = 1/2$, so that $s = 1/4$ lies strictly between $0$ and $s_i$. He proved that the data-to-solution map fails to be $C^5$ in $H^s$ for $s < s_m$, and norm inflation for $1 \le s < s_i$ when $s_i > 1$ and, for the one-dimensional cubic model, below $\min(s_i, 0)$. Kawakami and Murphy proved that every initial datum in $\{f \in H^1(\mathbb{R}^3) : xf \in L^2\}$ gives a unique global solution, which scatters.
Problem. For every $\varepsilon > 0$, are there $u_0 \in \mathcal{S}(\mathbb{R}^3)$ and a time $t_\varepsilon$ with $0 < |t_\varepsilon| < \varepsilon$, in the interval of existence of the smooth solution $u$ with $u(0) = u_0$, such that $$\|u_0\|_{\dot H^{1/4}} < \varepsilon, \qquad \|u(t_\varepsilon)\|_{\dot H^{1/4}} > \varepsilon^{-1}?$$
Source: Kowalski (2025), Definition 1.3 and Section 6.4: norm inflation in Ḣ^s, conjectured below the integrated-scaling index s_i and left unresolved in the intercritical case. Stated by AIM — Open Applied Problems for the defocusing quintic equation on R^3 at s = 1/4, with Schwartz data and a nonzero time.
Imported from github.com/MColbrook/AIM.
Classification
- Primary
- AP — Analysis of PDEs
- Secondary
- MP
- Keywords
- AIM Open Applied Problems, dispersion management, nonlinear Schrödinger equations, norm inflation
References
@misc{kowalski2025,
author = {Kowalski, Matthew},
title = {On ill-posedness for the {G}abitov--{T}uritsyn equation},
year = {2025},
eprint = {2510.11887},
archivePrefix = {arXiv},
}
@article{kawakami2026,
author = {Kawakami, Jumpei and Murphy, Jason},
title = {Small and large data scattering for the dispersion-managed {NLS}},
journal = {Discrete Contin. Dyn. Syst.},
volume = {47},
year = {2026},
pages = {256--285},
doi = {10.3934/dcds.2025118},
eprint = {2407.11151},
archivePrefix = {arXiv},
}
@misc{aim-intercritical-dispersion-managed-norm-inflation,
author = {Brunton, Steve and Colbrook, Matthew J. and de Hoop, Maarten V. and Stepaniants, George and Townsend, Alex and Ward, Rachel},
title = {{Intercritical norm inflation for the dispersion-managed Schrödinger equation}},
howpublished = {AIM --- Open Applied Problems, GitHub repository},
year = {2026},
url = {https://github.com/MColbrook/AIM/blob/24edaadc6bb272737d8da0745c662c992db07b22/problems/413-intercritical-dispersion-managed-norm-inflation.md},
note = {Entry 413 at commit 24edaad, 2026-10-15},
}
Resolution claims
We prove norm inflation in the homogeneous Sobolev space $\dot H^{1/4}(\R^3)$ for the defocusing quintic Gabitov--Turitsyn equation with averaging interval $[0,1]$. A family of Schwartz initial data has arbitrarily small initial norm, while its global smooth solutions attain arbitrarily large norms at arbitrarily small negative times. The proof combines a finite-interval virial identity with a lower bound for the averaged …
Status history
| When | Change | By | Reason |
|---|---|---|---|
| 2026-10-18 | Needs verification → Solution claimed | — | Recorded here: AIM @ 24edaad, paper/main.pdf. |
Citing this problem
This identifier is permanent. It will not change if the problem is solved, reclassified or withdrawn.
OP-00802, "Intercritical norm inflation for the dispersion-managed Schrödinger equation", clopen.solutions.