The stationary cubic nonlinear Dirac equation on the plane, with nonlinearity ∇Gβ1,β2\nabla G_{\beta_1,\beta_2} where Gβ1,β2(ψ)=β14(∣ψ1∣4+∣ψ2∣4)+β2∣ψ1∣2∣ψ2∣2G_{\beta_1,\beta_2}(\psi)=\tfrac{\beta_1}{4}(|\psi_1|^4+|\psi_2|^4)+\beta_2|\psi_1|^2|\psi_2|^2, is the effective model obtained by multiscale Bloch analysis of the nonlinear Schrödinger/Gross–Pitaevskii equation near the conical Dirac points of honeycomb-structured media. Borrelli (2018) constructed infinitely many weakly localized (non-square-integrable) stationary solutions of the massless equation within a rotational ansatz, characterized them as least-action critical points in that ansatz subspace, and posed one existence-theoretic question left open: (a) in the massive case, where the mass term breaks the scale invariance that produced the massless continuum, whether infinitely many distinct localized solitary waves exist for general coupling constants (β1,β2)(\beta_1,\beta_2). A second question is left open by the subsequent literature, though it was not posed as open in the 2018 paper (whose Remark 14 asserts that the variational argument works for H˙1/2\dot H^{1/2}-solutions of arbitrary form, while no existence statement outside the symmetric ansatz is proved there): (b) whether least-action ground states, or any stationary solutions of non-symmetric form, exist in the unrestricted energy space H˙1/2(R2,C2)\dot H^{1/2}(\mathbb{R}^2,\mathbb{C}^2). Subsequent work resolved the massive multiplicity question only for the pure cubic nonlinearity (β1=2β2\beta_1=2\beta_2) and only within symmetric vorticity classes or near spectral band edges. The open problem is to complete the existence and multiplicity theory of localized stationary solutions of this effective Dirac equation — massless and massive — for general positive couplings and without symmetry restrictions.

Background

Honeycomb-structured media — conduction electrons in graphene, light in photonic lattices, and Bose–Einstein condensates in optical lattices — are modeled at the linear level by a Schrödinger operator H=−Δ+V(x)H=-\Delta+V(x) whose potential VV has the symmetry of the hexagonal (honeycomb) lattice. Fefferman and Weinstein proved that for generic such potentials the dispersion bands of HH contain conical intersection points, called Dirac points, where two bands meet with a linear, cone-shaped dispersion law. Wave packets spectrally concentrated at a Dirac point are therefore governed, to leading order, not by a Schrödinger equation but by a two-component massless Dirac equation (Fefferman–Weinstein, 2014); with a cubic focusing nonlinearity present in the underlying nonlinear Schrödinger/Gross–Pitaevskii (NLS/GP) equation, a multiscale expansion of Bloch functions at the Dirac point produces an effective cubic nonlinear Dirac system for the slowly varying spinor amplitudes, with coupling constants β1,β2>0\beta_1,\beta_2>0 given by integrals of the Bloch modes over a lattice cell. This approximation is justified rigorously for the time-dependent (evolution) problem on finite time intervals (Arbunich–Sparber, 2018), but not for stationary states.

Stationary states of the effective system are described by the following equation. Let ψ=(ψ1,ψ2):R2→C2\psi=(\psi_1,\psi_2):\mathbb{R}^2\to\mathbb{C}^2 be a spinor, D=−i(σ1∂x1+σ2∂x2)\mathcal{D}=-i(\sigma_1\partial_{x_1}+\sigma_2\partial_{x_2}) the two-dimensional Dirac operator built from the Pauli matrices σ1,σ2\sigma_1,\sigma_2, and σ3\sigma_3 the third Pauli matrix. The massless equation (the stationary effective equation at the Dirac-point energy) is

Dψ=∇Gβ1,β2(ψ),Gβ1,β2(ψ)=β14(∣ψ1∣4+∣ψ2∣4)+β2∣ψ1∣2∣ψ2∣2,\mathcal{D}\psi = \nabla G_{\beta_1,\beta_2}(\psi), \qquad G_{\beta_1,\beta_2}(\psi)=\tfrac{\beta_1}{4}(|\psi_1|^4+|\psi_2|^4)+\beta_2|\psi_1|^2|\psi_2|^2,

where ∇G\nabla G is the gradient in the spinor variables, so ∇Gβ1,β2(ψ)=((β1∣ψ1∣2+2β2∣ψ2∣2)ψ1, (β1∣ψ2∣2+2β2∣ψ1∣2)ψ2)\nabla G_{\beta_1,\beta_2}(\psi)=\big((\beta_1|\psi_1|^2+2\beta_2|\psi_2|^2)\psi_1,\,(\beta_1|\psi_2|^2+2\beta_2|\psi_1|^2)\psi_2\big). When the honeycomb symmetry is perturbed so that a spectral gap opens at the Dirac point, a mass term appears and one obtains the massive equation

(D+mσ3−ω)ψ=∇Gβ1,β2(ψ),m>0,  ω∈(−m,m),(\mathcal{D}+m\sigma_3-\omega)\psi = \nabla G_{\beta_1,\beta_2}(\psi), \qquad m>0,\ \ \omega\in(-m,m),

where the frequency ω\omega must lie in the spectral gap (−m,m)(-m,m) of the free massive Dirac operator, whose spectrum is (−∞,−m]∪[m,+∞)(-\infty,-m]\cup[m,+\infty). In the pure cubic case β1=2β2\beta_1=2\beta_2 (after normalization β1=1\beta_1=1, β2=1/2\beta_2=1/2) one has ∇Gβ1,β2(ψ)∝∣ψ∣2ψ\nabla G_{\beta_1,\beta_2}(\psi)\propto|\psi|^2\psi.

The nonlinearity is critical for the Sobolev embedding: the natural energy space is the homogeneous Sobolev space H˙1/2(R2,C2)\dot H^{1/2}(\mathbb{R}^2,\mathbb{C}^2), in which the quadratic form ⟨Dψ,ψ⟩\langle\mathcal{D}\psi,\psi\rangle controls the H˙1/2\dot H^{1/2}-norm, and the embedding H˙1/2(R2)↪L4(R2)\dot H^{1/2}(\mathbb{R}^2)\hookrightarrow L^4(\mathbb{R}^2) is exactly borderline for the quartic energy. Two structural difficulties dominate the analysis. First, the action functional L(ψ)=12∫⟨Dmψ,ψ⟩ dx−∫G(ψ) dx\mathcal{L}(\psi)=\tfrac12\int\langle\mathcal{D}_m\psi,\psi\rangle\,dx-\int G(\psi)\,dx is strongly indefinite, because the Dirac operator has spectrum unbounded above and below, so direct minimization is impossible and minimax or duality schemes are required. Second, the massless equation is invariant under translations, phase rotations, and the scaling ψ(x)↦λ1/2ψ(λx)\psi(x)\mapsto\lambda^{1/2}\psi(\lambda x) (conformal scale invariance), so Palais–Smale sequences can lose compactness; the mass term breaks the scaling but not the translations.

Known results. Borrelli (2018) proved that for 0<β2≤β10<\beta_2\le\beta_1 the massless equation admits a smooth solution of the rotational (vorticity) ansatz ψ(r,θ)=(iu(r)eiθ,v(r))\psi(r,\theta)=(iu(r)e^{i\theta},v(r)) — unique up to scaling and sign, with exact asymptotics ∣ψ(r)∣∼C/r|\psi(r)|\sim C/r at infinity. These solutions are weakly localized: they lie in H˙1/2∩L4\dot H^{1/2}\cap L^4 but are not square-integrable, and scaling produces a continuum of them. Within the ansatz subspace they are least-action critical points (a ground-state-type variational characterization, proved via a dual functional and concentration–compactness, an approach introduced for such problems by Lions). For the massive case, the same ansatz with vorticity S=0S=0 yields, for every ω∈(−m,m)\omega\in(-m,m) and general couplings, a smooth exponentially localized solution (Borrelli 2017; Borrelli 2018, Theorem 24). Borrelli–Frank (2020) sharpened the massless picture: for every integer vorticity SS and general β1,β2>0\beta_1,\beta_2>0 there is a unique-up-to-symmetries ansatz solution — the S=0S=0 ground state and, for S≠0S\ne0, square-integrable excited states with faster polynomial decay and explicit asymptotics. All of these constructions live inside vorticity symmetry classes. Borrelli (2022) then proved, for the pure cubic massive equation only (β1=2β2\beta_1=2\beta_2), that for every S≠0S\ne0 and every ω∈(−m,m)\omega\in(-m,m) there is an exponentially localized solution — hence infinitely many distinct localized massive solitary waves — found variationally inside the symmetric subspaces, with the massless Borrelli–Frank profiles appearing as controlled blow-up limits. Borrelli–Carlone (2021) treated general β1,β2>0\beta_1,\beta_2>0 in the massive case, but only by bifurcation from the band edges ω→±m\omega\to\pm m and only within the radial ansatz. For comparison, in band-edge (gapped) regimes of periodic NLS the analogous effective-media approximation for standing waves is rigorous (Ilan–Weinstein, 2010); at the gapless Dirac point no such stationary approximation theorem is known.

Problem Statement

Consider the stationary cubic Dirac equation on R2\mathbb{R}^2 arising as the effective model for the nonlinear Schrödinger/Gross–Pitaevskii equation near Dirac points of honeycomb structures: the massless equation

Dψ=∇Gβ1,β2(ψ),Gβ1,β2(ψ)=β14(∣ψ1∣4+∣ψ2∣4)+β2∣ψ1∣2∣ψ2∣2,\mathcal{D}\psi = \nabla G_{\beta_1,\beta_2}(\psi), \qquad G_{\beta_1,\beta_2}(\psi)=\tfrac{\beta_1}{4}(|\psi_1|^4+|\psi_2|^4)+\beta_2|\psi_1|^2|\psi_2|^2,

and its massive (gap-opened) version

(D+mσ3−ω)ψ=∇Gβ1,β2(ψ),m>0,  ω∈(−m,m),(\mathcal{D}+m\sigma_3-\omega)\psi = \nabla G_{\beta_1,\beta_2}(\psi), \qquad m>0,\ \ \omega\in(-m,m),

with coupling constants β1,β2>0\beta_1,\beta_2>0 (the regime 0<β2≤β10<\beta_2\le\beta_1 is the one treated in the source paper; the question is posed for general positive couplings). Here ψ=(ψ1,ψ2)\psi=(\psi_1,\psi_2) is a two-component spinor; D=−i(σ1∂x1+σ2∂x2)\mathcal{D}=-i(\sigma_1\partial_{x_1}+\sigma_2\partial_{x_2}) is the two-dimensional massless Dirac operator, where σ1,σ2,σ3\sigma_1,\sigma_2,\sigma_3 are the standard 2×22\times2 Pauli matrices; and ∇Gβ1,β2(ψ)\nabla G_{\beta_1,\beta_2}(\psi) denotes the gradient of Gβ1,β2G_{\beta_1,\beta_2} in the spinor variables, i.e. ((β1∣ψ1∣2+2β2∣ψ2∣2)ψ1, (β1∣ψ2∣2+2β2∣ψ1∣2)ψ2)\big((\beta_1|\psi_1|^2+2\beta_2|\psi_2|^2)\psi_1,\,(\beta_1|\psi_2|^2+2\beta_2|\psi_1|^2)\psi_2\big). By a localized stationary solution we mean a nontrivial spinor of finite action, ψ∈H˙1/2(R2,C2)∩L4(R2,C2)\psi\in\dot H^{1/2}(\mathbb{R}^2,\mathbb{C}^2)\cap L^4(\mathbb{R}^2,\mathbb{C}^2), solving the equation weakly — including the weakly localized (non-L2L^2) solutions with algebraic decay in the massless case, and the exponentially decaying ones in the massive case.

The problem is to complete the existence and multiplicity theory of localized stationary solutions of this equation in the two regimes left open by the current literature:

(1) General nonlinearity, massive case. For arbitrary β1,β2>0\beta_1,\beta_2>0 — not only the pure cubic case β1=2β2\beta_1=2\beta_2 — determine whether, and for which frequencies ω∈(−m,m)\omega\in(-m,m), the massive equation admits infinitely many localized stationary solutions that are pairwise distinct (not related by the natural symmetries of the equation: translations, phase rotations, and the spin symmetries — spatial rotations and reflections acting on the spinor components, together with the sign symmetry ψ↦−ψ\psi\mapsto-\psi); in particular, whether the existence established for the pure cubic nonlinearity extends to general couplings at interior frequencies of the gap, beyond the known band-edge bifurcation and the vorticity-S=0S=0 construction, and whether a least-action (ground-state-type) solution exists in this generality.

(2) Beyond the symmetry ansatz. For the massless equation (and analogously for the massive one), determine whether localized stationary solutions exist that are not, up to the symmetries of the equation, of the vorticity-ansatz form ψ(r,θ)=(v(r)eiSθ, iu(r)ei(S+1)θ)\psi(r,\theta)=\big(v(r)e^{iS\theta},\,iu(r)e^{i(S+1)\theta}\big), S∈ZS\in\mathbb{Z} (the component ordering here and in the Background ansatz (iu(r)eiθ,v(r))(iu(r)e^{i\theta},v(r)) differ by the symmetry ψ(x)↦σ1ψ(x1,−x2)\psi(x)\mapsto\sigma_1\psi(x_1,-x_2) of the massless equation); in particular, whether the least-action ground state exists as a minimizer in the unrestricted space H˙1/2(R2,C2)\dot H^{1/2}(\mathbb{R}^2,\mathbb{C}^2), or conversely whether every localized stationary solution is of ansatz form (classification).

The two parts serve one objective — a complete inventory of the localized stationary states of the effective honeycomb Dirac equation — and neither implies the other; both are open. The question is posed for the stated equations with general positive couplings (β1,β2)(\beta_1,\beta_2), not only in the known special cases (pure cubic nonlinearity, symmetric vorticity subspaces, or band-edge frequencies).

References

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