Interacting helical vortex filaments in the 3-dimensional Ginzburg-Landau equation

Abstract

For each given $n\ge 2$, we construct a family of entire solutions $u_{\epsilon }(z,t)$, $\epsilon >0$, with helical symmetry to the 3-dimensional complex-valued Ginzburg-Landau equation \begin{equation*}\nonumber Δu+(1-|u|^2)u=0, \quad (z,t) \in \mathbb{R}^2\times \mathbb{R} \simeq \mathbb{R}^3. \end{equation*} These solutions are $2\pi /\epsilon$-periodic in $t$ and have $n$ helix-vortex curves, with asymptotic behavior as $\epsilon \to 0$ $$u_{\epsilon }(z,t)\approx \prod _{j=1}^{n}W\left(z-{\epsilon }^{-1}{f}_j(\epsilon t)\right),$$ where $W(z)=w(r)e^{i\theta }$, $z=r{e}^{i\theta },$ is the standard degree $+1$ vortex solution of the planar Ginzburg-Landau equation $\Delta W+(1-|W{|}^2)W=0\text{ in }{\mathbb{R}}^2$ and $$f_j(t)=\frac{\sqrt{n-1}{e}^{it}{e}^{2i(j-1)\pi /n}}{\sqrt{|\log \epsilon |}},j=1,\dots ,n.$$ Existence of these solutions was previously conjectured, being $\mathbf{f}(t)=(f_1(t),\dots ,f_n(t))$ a rotating equilibrium point for the renormalized energy of vortex filaments there derived, $${\mathcal{W}}_{\epsilon }(\mathbf{f}):=\pi {\int }_0^{2\pi }(\frac{|\log \epsilon |}{2}\sum _{k=1}^n|f_k^{\prime }(t){|}^2-\sum _{j\ne k}\log |f_j(t)-f_k(t)|)\mathrm{d}t,$$ corresponding to that of a planar logarithmic $n$-body problem. These solutions satisfy $$\underset{|z|\to +\infty }{\lim }|u_{\epsilon }(z,t)|=1uniformly\; int$$ and have nontrivial dependence on $t$, thus negatively answering the Ginzburg-Landau analogue of the Gibbons conjecture for the Allen-Cahn equation, a question originally formulated by H. Brezis.