Interacting helical vortex filaments in the three-dimensional Ginzburg–Landau equation

  • Juan Dávila

    University of Bath, UK
  • Manuel del Pino

    University of Bath, UK; Universidad de Chile, Santiago, Chile
  • Maria Medina

    Universidad Autónoma de Madrid, Spain
  • Rémy Rodiac

    Université Paris-Saclay, Orsay, France
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Abstract

For each given n2n\ge 2, we construct a family of entire solutions uε(z,t)u_\varepsilon (z,t), ε>0\varepsilon > 0, with helical symmetry to the three-dimensional complex-valued Ginzburg–Landau equation

Δu+(1u2)u=0,(z,t)R2×RR3.\Delta u+(1-|{u}|^2)u=0, \quad (z,t) \in \mathbb{R}^2\times \mathbb{R} \simeq \mathbb{R}^3.

These solutions are 2π/ε2\pi/\varepsilon-periodic in tt and have nn helix-vortex curves, with asymptotic behavior, as ε0\varepsilon\to 0,

uε(z,t)j=1nW(zε1fj(εt)),u_\varepsilon (z,t) \approx \mathop{\smash[t]{\prod_{j=1}^n}\vphantom{\prod}} W\bigl(z- \varepsilon^{-1} f_j(\varepsilon t)\bigr),

where W(z)=w(r)eiθW(z) =w(r) e^{i\theta}, z=reiθz= re^{i\theta}, is the standard degree +1+1 vortex solution of the planar Ginzburg–Landau equation ΔW+(1W2)W=0\Delta W+(1-|{W}|^2)W=0 in R2\mathbb{R}^2 and

fj(t)=n1eite2i(j1)π/nlogε,j=1,,n.f_j(t) = \frac{\sqrt{n-1} e^{it}e^{2 i (j-1)\pi/ n}}{\sqrt{|{\log\varepsilon}|}}, \quad j=1,\ldots, n.

Existence of these solutions was previously conjectured by del Pino and Kowalczyk (2008), f(t)=(f1(t),,fn(t))\mathbf{f}(t) = (f_1(t),\ldots, f_n(t)) being a rotating equilibrium point for the renormalized energy of vortex filaments derived there,

Wε(f):=π02π(logε2k=1nfk(t)2jklogfj(t)fk(t))dt,\mathcal{W}_\varepsilon (\mathbf{f}) := \pi \int_0^{2\pi} \Biggl( \frac{|{\log\varepsilon}|} 2 \sum_{k=1}^n |{f'_k(t)}|^2 - \sum_{j\neq k}\log |{f_j(t)-f_k(t)}| \Biggr) \operatorname{d}t,

corresponding to that of a planar logarithmic nn-body problem. The modulus of these solutions converges to 11 as z|{z}| goes to infinity uniformly in (t)(t), and the solutions have nontrivial dependence on tt, thus negatively answering the Ginzburg–Landau analogue of the Gibbons conjecture for the Allen–Cahn equation, a question originally formulated by H. Brezis.

Cite this article

Juan Dávila, Manuel del Pino, Maria Medina, Rémy Rodiac, Interacting helical vortex filaments in the three-dimensional Ginzburg–Landau equation. J. Eur. Math. Soc. 24 (2022), no. 12, pp. 4143–4199

DOI 10.4171/JEMS/1175