Radial classification of higher-degree entire planar Ginzburg–Landau vortices
Let $u \in C^\infty(\mathbb{R}^2; \mathbb{C})$ solve $$-\Delta u = u(1 - |u|^2) \quad \text{in } \mathbb{R}^2, \qquad \lim_{|x| \to \infty} |u(x)| = 1,$$ and let its degree be the degree of $u/|u|$ on the circles $|x| = R$ for sufficiently large $R$. For each integer $q \ne 0$ the problem $$g'' + \frac{g'}{r} - \frac{q^2}{r^2}\, g + g(1 - g^2) = 0 \ \text{ on } (0, \infty), \qquad g(0) = 0, \quad \lim_{r \to \infty} g(r) = 1,$$ has a unique solution $g_q$, with values in $[0, 1)$, and $V_q(re^{i\theta}) = e^{iq\theta} g_q(r)$ is a solution of degree $q$. Brezis, Merle and Rivière proved that a solution of degree $0$ with $\int_{\mathbb{R}^2} (|u|^2 - 1)^2 < \infty$ is a constant, and Mironescu that a solution of degree $\pm 1$ with this property is $V_{\pm 1}$ up to a translation and a unimodular factor. Ovchinnikov and Sigal devised a strategy to construct nonradial solutions of degree at least two in absolute value, based on a correlation term between distant vortices; Esposito proved that the correlation coefficient they introduced always vanishes.
Problem. Let $q \ge 2$ and let $u$ be such a solution of degree $q$. Must there be $x_0 \in \mathbb{R}^2$ and $\alpha \in \mathbb{C}$ with $|\alpha| = 1$ such that $$u(x_0 + re^{i\theta}) = \alpha\, e^{iq\theta} g_q(r) \qquad (r \ge 0,\ \theta \in \mathbb{R})?$$
Source: Bethuel, Brezis and Hélein (1994), Problem 14; see Brezis (2023), Open Problem 2.4, for every degree. Stated for degrees q ≥ 2 by AIM — Open Applied Problems.
Imported from github.com/MColbrook/AIM.
Classification
- Primary
- AP — Analysis of PDEs
- Secondary
- MP
- Keywords
- AIM Open Applied Problems, Ginzburg–Landau equation, radial symmetry, vortices
References
@book{bethuel1994,
author = {Bethuel, Fabrice and Brezis, Ha{\"i}m and H{\'e}lein, Fr{\'e}d{\'e}ric},
title = {Ginzburg--{L}andau Vortices},
series = {Progress in Nonlinear Differential Equations and their Applications},
volume = {13},
publisher = {Birkh{\"a}user},
address = {Boston, MA},
year = {1994},
doi = {10.1007/978-1-4612-0287-5},
}
@article{brezis2023,
author = {Brezis, Ha{\"i}m},
title = {Some of my favorite open problems},
journal = {Rend. Lincei Mat. Appl.},
volume = {34},
number = {2},
year = {2023},
pages = {307--335},
doi = {10.4171/RLM/1008},
}
@article{brezis1994,
author = {Brezis, Ha{\"i}m and Merle, Frank and Rivi{\`e}re, Tristan},
title = {Quantization effects for {$-\Delta u = u(1 - |u|^2)$} in {$\mathbb{R}^2$}},
journal = {Arch. Rational Mech. Anal.},
volume = {126},
number = {1},
year = {1994},
pages = {35--58},
doi = {10.1007/BF00375695},
}
@article{mironescu1996,
author = {Mironescu, Petru},
title = {Les minimiseurs locaux pour l'{\'e}quation de {G}inzburg--{L}andau sont {\`a} sym{\'e}trie radiale},
journal = {C. R. Acad. Sci. Paris S{\'e}r. I Math.},
volume = {323},
number = {6},
year = {1996},
pages = {593--598},
}
@article{ovchinnikov2004,
author = {Ovchinnikov, Yu. N. and Sigal, I. M.},
title = {Symmetry-breaking solutions of the {G}inzburg--{L}andau equation},
journal = {J. Exp. Theor. Phys.},
volume = {99},
number = {5},
year = {2004},
pages = {1090--1107},
doi = {10.1134/1.1842890},
}
@article{esposito2013,
author = {Esposito, Pierpaolo},
title = {Some remarks concerning symmetry-breaking for the {G}inzburg--{L}andau equation},
journal = {J. Funct. Anal.},
volume = {265},
number = {10},
year = {2013},
pages = {2189--2203},
doi = {10.1016/j.jfa.2013.07.029},
eprint = {1210.6315},
archivePrefix = {arXiv},
}
@misc{aim-entire-ginzburg-landau-vortex-symmetry,
author = {Brunton, Steve and Colbrook, Matthew J. and de Hoop, Maarten V. and Stepaniants, George and Townsend, Alex and Ward, Rachel},
title = {{Radial classification of higher-degree entire planar Ginzburg–Landau vortices}},
howpublished = {AIM --- Open Applied Problems, GitHub repository},
year = {2026},
url = {https://github.com/MColbrook/AIM/blob/24edaadc6bb272737d8da0745c662c992db07b22/problems/436-entire-ginzburg-landau-vortex-symmetry.md},
note = {Entry 436 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-00858, "Radial classification of higher-degree entire planar Ginzburg–Landau vortices", clopen.solutions.