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

This is the authors' abstract. We don't add key points for this paper.

For each given n≥ 2 , we construct a family of entire solutions u_ε (z,t) , ε > 0 , with helical symmetry to the three-dimensional complex-valued Ginzburg–Landau equation Δ u+(1-|{u}|^2)u=0, (z,t) ∈ R^2× R R^3. These solutions are 2π/ε -periodic in t and have n helix-vortex curves, with asymptotic behavior, as ε→ 0 , u_ε (z,t) ≈ ∏ⱼ₌₁^n W(z- ε⁻¹ f_j(ε t)), where W(z) =w(r) eiθ , z= reiθ , is the standard degree +1 vortex solution of the planar Ginzburg–Landau equation Δ W+(1-|{W}|^2)W=0 in R^2 and f_j(t) = {√n-1 eⁱᵗe2 i (j-1)π/ n}{√{|{logε}|}}, j=1,…, n. Existence of these solutions was previously conjectured by del Pino and Kowalczyk (2008), f(t) = (f_1(t),…, f_n(t)) being a rotating equilibrium point for the renormalized energy of vortex filaments derived there, W_ε (f) := π ∫_02π ( {|{logε}|} 2 ∑ₖ₌₁^n |{f'_k(t)}|^2 - ∑j≠ klog |{f_j(t)-f_k(t)}| ) dt, corresponding to that of a planar logarithmic n -body problem. The modulus of these solutions converges to 1 as |{z}| goes to infinity uniformly in (t) , and the solutions 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.