三维Ginzburg-Landau方程中相互作用的螺旋涡丝
Interacting helical vortex filaments in the 3-dimensional Ginzburg-Landau equation
摘要：
对于每个给定的$n\geq 2$，我们构造了三维复值Ginzburg-Landau方程\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*}的一族螺旋对称的整解$u_\varepsilon (z,t)$，$\varepsilon>0$。这些解在$t$中是$2π/\varepsilon$ -周期的，并且具有$n$螺旋涡旋曲线，渐近行为为$\varepsilon\to 0$ $$ u_\varepsilon (z,t) \approx \prod_{j=1}^n W\left( z- \varepsilon^{-1} f_j(\varepsilon t) \right), $$，其中$W(z) =w(r) e^{iθ} $，$z= re^{iθ},$是平面Ginzburg-Landau方程$ ΔW+(1-|W|^2)W=0 \text{ in } \mathbb{R}^2 $和$$ f_j(t) = \frac { \sqrt{n-1} e^{it}e^{2 i (j-1)π/ n }}{ \sqrt{|\log\varepsilon|}}, \quad j=1,\ldots, n. $$的标准阶$+1$涡解。由于${\bf f}(t) = (f_1(t),\ldots, f_n(t))$是涡丝重整化能量的旋转平衡点，因此导出了$$ \mathcal W_\varepsilon ( {\bf f} ) :=π\int_0^{2π} \Big ( \, \frac{|\log \varepsilon|} 2 \sum_{k=1}^n|f'_k(t)|^2-\sum_{j\neq k}\log |f_j(t)-f_k(t)| \, \Big ) \mathrm{d} t, $$对应于平面对数$n$ -体问题。这些解满足$$ \lim_{|z| \to +\infty } |u_\varepsilon (z,t)| = 1 \quad \hbox{uniformly in $t$} $$，并且对$t$有非平凡的依赖性，从而否定地回答了最初由H.布雷齐斯
英文摘要：
For each given $n\geq 2$, we construct a family of entire solutions $u_\varepsilon (z,t)$, $\varepsilon>0$, 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 $2π/\varepsilon$-periodic in $t$ and have $n$ helix-vortex curves, with asymptotic behavior as $\varepsilon\to 0$ $$ u_\varepsilon (z,t) \approx \prod_{j=1}^n W\left( z- \varepsilon^{-1} f_j(\varepsilon t) \right), $$ where $W(z) =w(r) e^{iθ} $, $z= re^{iθ},$ is the standard degree $+1$ vortex solution of the planar Ginzburg-Landau equation $ ΔW+(1-|W|^2)W=0 \text{ in } \mathbb{R}^2 $ and $$ f_j(t) = \frac { \sqrt{n-1} e^{it}e^{2 i (j-1)π/ n }}{ \sqrt{|\log\varepsilon|}}, \quad j=1,\ldots, n. $$ Existence of these solutions was previously conjectured, being ${\bf f}(t) = (f_1(t),\ldots, f_n(t))$ a rotating equilibrium point for the renormalized energy of vortex filaments there derived, $$ \mathcal W_\varepsilon ( {\bf f} ) :=π\int_0^{2π} \Big ( \, \frac{|\log \varepsilon|} 2 \sum_{k=1}^n|f'_k(t)|^2-\sum_{j\neq k}\log |f_j(t)-f_k(t)| \, \Big ) \mathrm{d} t, $$ corresponding to that of a planar logarithmic $n$-body problem. These solutions satisfy $$ \lim_{|z| \to +\infty } |u_\varepsilon (z,t)| = 1 \quad \hbox{uniformly in $t$} $$ 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.
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