Uniqueness of the zero-average dispersion-managed soliton
For $r \in \mathbb{R}$ let $T_r = e^{ir\partial_x^2}$ be the free Schrödinger evolution on $L^2(\mathbb{R}; \mathbb{C})$, and let $$\mathcal{Q}(f) = \int_0^1 \int_{\mathbb{R}} |T_r f(x)|^4 \, dx \, dr, \qquad P = \sup_{\|f\|_2 = 1} \mathcal{Q}(f).$$ Maximizers of this problem give the dispersion-managed solitons with zero average dispersion, for the dispersion profile whose distribution is uniform on $[0, 1]$. Kunze proved that $P$ is attained, and Hundertmark and Lee gave a direct proof of existence that covers more general profiles. Erdoğan, Hundertmark and Lee proved that every maximizer and its Fourier transform decay exponentially. For a related minimization problem with positive average dispersion and a parameter, Choi, Jin and Lee proved that the minimizer is unique up to shifts and boosts for small values of the parameter.
Let $G$ be the group generated by the phase rotations $f \mapsto e^{i\theta} f$, the translations $f \mapsto f(\cdot - a)$, the modulations $f(x) \mapsto e^{i\xi x} f(x)$, the reflection $f(x) \mapsto f(-x)$ and the dispersion-time reversal $f \mapsto T_{-1} \overline{f}$, with $\theta, a, \xi \in \mathbb{R}$. Each of them preserves $\mathcal{Q}$ and the $L^2$ norm.
Problem. Are any two maximizers of $P$ in the same $G$-orbit? Only maximizers are concerned, not arbitrary critical points of $\mathcal{Q}$ on the unit sphere of $L^2(\mathbb{R})$.
Source: Hundertmark and Schnaubelt, Project B2 (Dispersion Management, 2015–2019) of the Collaborative Research Centre 1173 at KIT, as uniqueness of the ground states modulo natural invariances, for the variational problem of Kunze (2004) and Hundertmark and Lee (2012). Stated in this form, with reflection and dispersion-time reversal among the symmetries, by AIM — Open Applied Problems.
Imported from github.com/MColbrook/AIM.
Classification
- Primary
- AP — Analysis of PDEs
- Secondary
- MP
- Keywords
- AIM Open Applied Problems, dispersion management, ground states, nonlinear Schrödinger equations
References
@misc{kit2015b2,
author = {Hundertmark, Dirk and Schnaubelt, Roland},
title = {Project {B2}: Dispersion Management},
howpublished = {Collaborative Research Centre 1173 ``Wave phenomena'', Karlsruhe Institute of Technology, project description},
year = {2015},
url = {https://www.waves.kit.edu/B2.php},
note = {Project period July 2015 to June 2019},
}
@article{kunze2004,
author = {Kunze, Markus},
title = {On a variational problem with lack of compactness related to the {S}trichartz inequality},
journal = {Calc. Var. Partial Differential Equations},
volume = {19},
number = {3},
year = {2004},
pages = {307--336},
doi = {10.1007/s00526-003-0218-9},
}
@article{hundertmark2012,
author = {Hundertmark, Dirk and Lee, Young-Ran},
title = {On non-local variational problems with lack of compactness related to non-linear optics},
journal = {J. Nonlinear Sci.},
volume = {22},
number = {1},
year = {2012},
pages = {1--38},
doi = {10.1007/s00332-011-9106-1},
eprint = {1008.4631},
archivePrefix = {arXiv},
}
@article{erdogan2011,
author = {Erdo{\u g}an, M. Burak and Hundertmark, Dirk and Lee, Young-Ran},
title = {Exponential decay of dispersion managed solitons for vanishing average dispersion},
journal = {Math. Res. Lett.},
volume = {18},
number = {1},
year = {2011},
pages = {11--24},
doi = {10.4310/MRL.2011.v18.n1.a2},
eprint = {1009.3707},
archivePrefix = {arXiv},
}
@article{choi2023,
author = {Choi, Mi-Ran and Jin, Sangdon and Lee, Young-Ran},
title = {Uniqueness of a minimizer for a variational problem related to the dispersion managed nonlinear {S}chr{\"o}dinger equation},
journal = {SIAM J. Math. Anal.},
volume = {55},
number = {4},
year = {2023},
pages = {2595--2614},
doi = {10.1137/22M1499972},
}
@misc{aim-zero-dispersion-soliton-uniqueness,
author = {Brunton, Steve and Colbrook, Matthew J. and de Hoop, Maarten V. and Stepaniants, George and Townsend, Alex and Ward, Rachel},
title = {{Uniqueness of the zero-average dispersion-managed soliton}},
howpublished = {AIM --- Open Applied Problems, GitHub repository},
year = {2026},
url = {https://github.com/MColbrook/AIM/blob/24edaadc6bb272737d8da0745c662c992db07b22/problems/418-zero-dispersion-soliton-uniqueness.md},
note = {Entry 418 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-00916, "Uniqueness of the zero-average dispersion-managed soliton", clopen.solutions.