De Giorgi-type one-dimensional symmetry for the Rabi-coupled fractional Gross–Pitaevskii system beyond the low-dimensional regimes

The classical De Giorgi conjecture predicts that bounded entire solutions of the Allen–Cahn equation that are monotone in one direction are one-dimensional in low dimensions. Wu and Zhang (2025) extended this program to the two-component, Rabi-coupled fractional Gross–Pitaevskii system that models coherently coupled Bose–Einstein condensates with long-range interactions, proving that positive bounded monotone entire solutions are one-dimensional in $\mathbb{R}^3$ for fractional exponent $s\in[1/2,1)$ and in $\mathbb{R}^2$ for $s\in(0,1/2)$. The completion of the classification remains open: one-dimensional symmetry is unsettled for $s\in(0,1/2)$ in dimensions $n\ge 3$ and for $s\in[1/2,1)$ in dimensions $n\ge 4$, and no counterexample (a monotone, non-one-dimensional solution) is known for any nonlocal version of the problem.

Background

A two-component Bose–Einstein condensate (BEC) is an ultracold gas in which two internal hyperfine states coexist; its mean-field description is given by coupled Gross–Pitaevskii equations for the two wave functions $u,v$. When the two states are coupled by a coherent laser field (Rabi coupling, with Rabi frequency $\omega$) and interact with each other (interaction parameter $\alpha>0$), the stationary model takes the form

$(-\Delta)^{s}u+u(u^{2}+v^{2}-1)+v(\alpha uv-\omega)=0,\qquad (-\Delta)^{s}v+v(u^{2}+v^{2}-1)+u(\alpha uv-\omega)=0\quad\text{in }\mathbb{R}^{n},\quad u,v>0,$

where $(-\Delta)^{s}$, $0<s<1$, is the fractional Laplacian, a nonlocal operator defined for smooth $u$ by the principal-value integral $(-\Delta)^{s}u(x)=C_{n,s}\,\mathrm{PV}\int_{\mathbb{R}^{n}}\frac{u(x)-u(y)}{|x-y|^{n+2s}}\,dy$; it interpolates between the identity ($s\to 0$) and $-\Delta$ ($s\to 1$) and models long-range interactions. The case $s=1$ recovers the local Rabi-coupled Gross–Pitaevskii system. For parameters $0<\omega<\alpha/2$ the system has, besides a symmetric constant state, two segregated constant equilibria $(a,b)$ and $(b,a)$ with $a,b>0$, $a^{2}+b^{2}=1$, $ab=\omega/\alpha$, and the physically interesting solutions are transition layers connecting $(a,b)$ to $(b,a)$ across an interface. A solution is called one-dimensional if $(u,v)(x)$ depends on a single Euclidean variable $x\cdot\nu$, and monotone if $\partial_{x_{n}}u>0>\partial_{x_{n}}v$ everywhere.

The prototype is the scalar Allen–Cahn equation $\Delta u=u^{3}-u$, the Euler–Lagrange equation of the Ginzburg–Landau energy with double-well potential. De Giorgi (1979) conjectured that bounded entire solutions of the Allen–Cahn equation that are monotone in one direction are one-dimensional, at least in low dimensions. This was proved in $\mathbb{R}^{2}$ by Ghoussoub–Gui (1998) and in $\mathbb{R}^{3}$ by Ambrosio–Cabré (2000); Savin (2009) proved it up to $n\le 8$ under the additional assumption that one level set is a complete graph; and del Pino–Kowalczyk–Wei (2011) constructed a non-one-dimensional monotone solution in $\mathbb{R}^{9}$, built on the Simons cone (the simplest non-flat minimal cone, which first exists in dimension 8). The intermediate range $4\le n\le 8$ remains open without extra hypotheses.

For the fractional Allen–Cahn equation $(-\Delta)^{s}u=u-u^{3}$ the analogous conjecture has a subtler structure: Savin–Valdinoci (2012) showed that the $\Gamma$-limit of the rescaled energy is the classical perimeter for $s\in[1/2,1)$ but a nonlocal perimeter (nonlocal minimal surfaces) for $s\in(0,1/2)$, so the geometric mechanism changes at $s=1/2$. The settled scalar cases are: $\mathbb{R}^{2}$ for all $s\in(0,1)$ (Sire–Valdinoci 2009; Cabré–Sire 2015; a new proof by Dipierro–Gonçalves da Silva–Poggesi–Valdinoci 2025); $\mathbb{R}^{3}$ for all $s$, namely $s=1/2$ by Cabré–Cinti (2010), $s\in(1/2,1)$ by Cabré–Cinti (2014), and $s\in(0,1/2)$ by Dipierro–Farina–Valdinoci (2018) and Dipierro–Serra–Valdinoci (2020); and $\mathbb{R}^{4}$ only for $s=1/2$ and stable solutions (Figalli–Serra 2020, working in the extension variables). Savin (2018, 2019) proved one-dimensional symmetry in $n\le 8$ for $s\ge 1/2$ under asymptotic flatness of the level sets, and Dipierro–Serra–Valdinoci (2020) proved an improvement-of-flatness theorem for all $s\in(0,1)$ under the same flatness assumption. All remaining regimes are open, and, remarkably, no counterexample is known for any fractional (nonlocal) version, in any dimension. A key technical tool throughout is the Caffarelli–Silvestre extension, which realizes $(-\Delta)^{s}$ as the Dirichlet-to-Neumann map of a degenerate local elliptic problem for an extended function $U(x,y)$ in the upper half-space $\mathbb{R}^{n+1}_{+}$ with weight $y^{1-2s}$; the energy estimates and Liouville arguments then operate in the extended space, where monotone solutions are treated by sliding and Hamiltonian-type identities.

For coupled systems the literature is younger. Fazly–Sire (2015) proved De Giorgi-type results for fractional gradient systems in $\mathbb{R}^{2}$ and $\mathbb{R}^{3}$ for certain exponents, under orientability and $H$-monotonicity hypotheses that the Rabi-coupled system does not satisfy (the system is non-cooperative, so no maximum principle holds for the pair). Aftalion–Farina–Nguyen (2021) proved the Gibbons-type version for the local Rabi-coupled system — one-dimensional symmetry in all dimensions under prescribed uniform limits at infinity — leaving the purely monotone (De Giorgi-type) question in higher dimensions open even for $s=1$. Wu–Zhang (paper I, arXiv:2509.19124) proved the local monotone result in $\mathbb{R}^{3}$; they also observed that at $\alpha=2$ the system decouples ($u+v$ constant, $u-v$ a rescaled Allen–Cahn solution), an algebraic fact valid verbatim for the fractional operator, so the coupled problem in any regime $(s,n)$ is at least as hard as the scalar problem there. Paper II (the source of this problem, arXiv:2511.16907) initiated the nonlocal theory: by adapting the Ambrosio–Cabré strategy through the Caffarelli–Silvestre extension, with a coupled Ginzburg–Landau energy and a Schrödinger-type linearized system, they proved that every positive, globally bounded, monotone entire solution of the fractional system is one-dimensional, with limits $(a,b)$ and $(b,a)$, in $\mathbb{R}^{3}$ for $s\in[1/2,1)$ and in $\mathbb{R}^{2}$ for $s\in(0,1/2)$. (The regimes established there are $n=3$ for $s\in[1/2,1)$ and $n=2$ for $s\in(0,1/2)$. Since the fractional Laplacian commutes with extension by constants in additional variables, the $\mathbb{R}^{3}$ result also covers $\mathbb{R}^{2}$ for $s\ge 1/2$, but no argument reduces higher dimensions to these.)

Problem Statement

For $s\in(0,1)$, $\alpha>0$, and $0<\omega<\alpha/2$, consider the Rabi-coupled fractional Gross–Pitaevskii system

$(-\Delta)^{s}u+u(u^{2}+v^{2}-1)+v(\alpha uv-\omega)=0,\qquad (-\Delta)^{s}v+v(u^{2}+v^{2}-1)+u(\alpha uv-\omega)=0\quad\text{in }\mathbb{R}^{n},$

with $u,v>0$ and globally bounded. Call $(u,v)$ monotone if $\partial_{x_{n}}u>0>\partial_{x_{n}}v$ in all of $\mathbb{R}^{n}$, and one-dimensional if $(u,v)(x)\equiv(U(x\cdot\nu),V(x\cdot\nu))$ for a unit vector $\nu$ and a one-dimensional profile with $U'>0>V'$. Wu–Zhang (2025) established one-dimensional symmetry of every positive, bounded, monotone entire solution in the regimes $s\in[1/2,1)$ with $n=3$ and $s\in(0,1/2)$ with $n=2$. The problem is to complete this De Giorgi-type classification for the remaining regimes:

(1) Prove or disprove that every positive, globally bounded, monotone entire solution is one-dimensional when $s\in(0,1/2)$ and $n\ge3$, and when $s\in[1/2,1)$ and $n\ge4$.

(2) Determine whether non-one-dimensional monotone entire solutions exist in high dimensions, and if so construct them; identify the critical dimension, and its dependence on the fractional exponent $s$, at which one-dimensional symmetry breaks down.

For every pair $(s,n)$ with $0<s<1$ and $n\ge1$, determine whether all such solutions are one-dimensional; a non-rigid regime is settled by exhibiting a monotone solution that is not one-dimensional. The single overarching objective is the classification of monotone entire solutions of this coupled nonlocal system, exactly as in the classical De Giorgi program. Part (2) is not independent of part (1): its existence question is the negative direction of part (1), so a proof or a counterexample resolving part (1) already settles whether non-one-dimensional monotone solutions exist in the regimes it covers, and exhibiting such a solution in a given regime answers part (1) there in the negative. Determine the critical dimension, if any, at which one-dimensional symmetry breaks down, and its dependence on the fractional exponent $s$.

References

1. Leyun Wu and Chilin Zhang, Phase transitions in two-component Bose-Einstein condensates with Rabi frequency (II): The De Giorgi conjecture for the nonlocal problem in ℝ² or ℝ³ (2025), arXiv:2511.16907, DOI 10.48550/arXiv.2511.16907, https://arxiv.org/abs/2511.16907 (source of this problem statement).
2. Leyun Wu and Chilin Zhang, Phase transitions in two-component Bose-Einstein condensates with Rabi frequency (I): The De Giorgi conjecture for the local problem in ℝ³ (2025), arXiv:2509.19124, DOI 10.48550/arXiv.2509.19124, https://arxiv.org/abs/2509.19124.
3. Xavier Cabré and Eleonora Cinti, Energy estimates and 1-D symmetry for nonlinear equations involving the half-Laplacian, Discrete and Continuous Dynamical Systems 28 (2010), 1179–1206, DOI 10.3934/dcds.2010.28.1179, https://doi.org/10.3934/dcds.2010.28.1179.
4. Xavier Cabré and Eleonora Cinti, Sharp energy estimates for nonlinear fractional diffusion equations, Calculus of Variations and Partial Differential Equations 49 (2014), 233–269, DOI 10.1007/s00526-012-0580-6, arXiv:1207.6194, https://arxiv.org/abs/1207.6194.
5. Serena Dipierro, Alberto Farina, and Enrico Valdinoci, A three-dimensional symmetry result for a phase transition equation in the genuinely nonlocal regime, Calculus of Variations and Partial Differential Equations 57 (2018), Paper No. 15, DOI 10.1007/s00526-017-1295-5, https://doi.org/10.1007/s00526-017-1295-5.
6. Serena Dipierro, Joaquim Serra, and Enrico Valdinoci, Improvement of flatness for nonlocal phase transitions, American Journal of Mathematics 142 (2020), 1083–1160, DOI 10.1353/ajm.2020.0032, arXiv:1611.10105, https://arxiv.org/abs/1611.10105.
7. Alessio Figalli and Joaquim Serra, On stable solutions for boundary reactions: a De Giorgi-type result in dimension 4+1, Inventiones mathematicae 219 (2020), 153–177, DOI 10.1007/s00222-019-00904-2, https://doi.org/10.1007/s00222-019-00904-2.
8. Ovidiu Savin, Rigidity of minimizers in nonlocal phase transitions, Analysis & PDE 11 (2018), 1881–1900, DOI 10.2140/apde.2018.11.1881, https://doi.org/10.2140/apde.2018.11.1881.
9. Ovidiu Savin, Rigidity of minimizers in nonlocal phase transitions II, Analysis in Theory and Applications 35 (2019), 1–27, DOI 10.4208/ata.oa-0008, https://doi.org/10.4208/ata.oa-0008.
10. Ovidiu Savin and Enrico Valdinoci, Γ-convergence for nonlocal phase transitions, Annales de l'Institut Henri Poincaré C, Analyse Non Linéaire 29 (2012), 479–500, DOI 10.1016/j.anihpc.2012.01.006, https://doi.org/10.1016/j.anihpc.2012.01.006.
11. Serena Dipierro, João Gonçalves da Silva, Giorgio Poggesi, and Enrico Valdinoci, Fractional De Giorgi conjecture in dimension 2 via complex-plane methods (2025), arXiv:2503.06082, DOI 10.48550/arXiv.2503.06082, https://arxiv.org/abs/2503.06082.
12. Mostafa Fazly and Yannick Sire, Symmetry results for fractional elliptic systems and related problems, Communications in Partial Differential Equations 40 (2015), 1070–1095, DOI 10.1080/03605302.2014.1003073, arXiv:1402.1193, https://arxiv.org/abs/1402.1193.
13. Amandine Aftalion, Alberto Farina, and Luc Nguyen, Moving planes for domain walls in a coupled system, Communications in Partial Differential Equations 46 (2021), 1410–1439, DOI 10.1080/03605302.2021.1881112, arXiv:1907.09777, https://arxiv.org/abs/1907.09777.
14. Alberto Farina and Enrico Valdinoci, Rigidity results for elliptic PDEs with uniform limits: an abstract framework with applications, Indiana University Mathematics Journal 60 (2011), 121–142, DOI 10.1512/iumj.2011.60.4433, https://doi.org/10.1512/iumj.2011.60.4433.
15. Ovidiu Savin, Regularity of flat level sets in phase transitions, Annals of Mathematics (2) 169 (2009), 41–78, DOI 10.4007/annals.2009.169.41, https://doi.org/10.4007/annals.2009.169.41.
16. Yannick Sire and Enrico Valdinoci, Fractional Laplacian phase transitions and boundary reactions: a geometric inequality and a symmetry result, Journal of Functional Analysis 256 (2009), 1842–1864, DOI 10.1016/j.jfa.2009.01.020, https://doi.org/10.1016/j.jfa.2009.01.020.
17. Xavier Cabré and Yannick Sire, Nonlinear equations for fractional Laplacians II: Existence, uniqueness, and qualitative properties of solutions, Transactions of the American Mathematical Society 367 (2015), 911–941, DOI 10.1090/s0002-9947-2014-05906-0, https://doi.org/10.1090/s0002-9947-2014-05906-0.
18. Manuel del Pino, Michał Kowalczyk, and Juncheng Wei, On De Giorgi's conjecture in dimension N ≥ 9, Annals of Mathematics (2) 174 (2011), 1485–1569, DOI 10.4007/annals.2011.174.3.3, https://doi.org/10.4007/annals.2011.174.3.3.
19. Luigi Ambrosio and Xavier Cabré, Entire solutions of semilinear elliptic equations in ℝ³ and a conjecture of De Giorgi, Journal of the American Mathematical Society 13 (2000), 725–739, DOI 10.1090/s0894-0347-00-00345-3, https://doi.org/10.1090/s0894-0347-00-00345-3.
20. Nassif Ghoussoub and Changfeng Gui, On a conjecture of De Giorgi and some related problems, Mathematische Annalen 311 (1998), 481–491, DOI 10.1007/s002080050196, https://doi.org/10.1007/s002080050196.