论文标题
Ginzburg-Landau系统的螺旋涡旋细丝$ {\ Mathbb r}^3 $
The helical vortex filaments of Ginzburg-Landau system in ${\mathbb R}^3$
论文作者
论文摘要
我们在$ {\ mathbb r}^3 $ \ begin {align*} \ begin {cases}-ε^2Δw^^+ +\ big [a _+\ big（| w^+|^2- {t^+}^2 \ big）+b \ big（| w^ - |^2- {t^ - }^2 \ big）\ big] w^+= 0，\\ [3mm] +\ big [a _- \ big（| w^ - |^2- {t^ - }^2 \ big）+b \ big（| w^+|^2- {t^+}^2 \ big）常数系数满足$$ a_+，a _-> 0，\ Quad B^2 <a_+a_--，\ Quad T^\ pm> 0，\ Quad {t^+}^2+ {t^ - }^ - }^2 = 1。 $$如果$ b <0 $，那么对于每一个$ε$，我们就会建立一个整个解决方案的家庭$w_ε（\ tilde {z}，t），t）\ in \ mathbb {c}^2 $ in canlindrical coortinates $（\ tilde {z}，tilde {z}，t）通过J.Dávila，M。DelPino，M。Medina和R. Rodiac在{\ tt arXiv：1901.02807}中引入的方法。这些解决方案为$2π$ - 周期为$ t $，并具有多个相互作用的涡旋螺旋。主要的结果是在$ \ Mathbb {r}^3 $中相互作用的（单个）Ginzburg-landau方程相互作用的螺旋涡旋丝的现象的扩展。我们的结果对Gibbons的猜想\ Cite {Gibbons猜想}对Ginzburg-Landau系统版本中的Allen-Cahn方程式进行了负面影响，这是H. Brezis最初提出的问题的扩展。
We consider the following coupled Ginzburg-Landau system in ${\mathbb R}^3$ \begin{align*} \begin{cases} -ε^2 Δw^+ +\Big[A_+\big(|w^+|^2-{t^+}^2\big)+B\big(|w^-|^2-{t^-}^2\big)\Big]w^+=0, \\[3mm] -ε^2 Δw^- +\Big[A_-\big(|w^-|^2-{t^-}^2\big)+B\big(|w^+|^2-{t^+}^2\big)\Big]w^-=0, \end{cases} \end{align*} where $w=(w^+, w^-)\in \mathbb{C}^2$ and the constant coefficients satisfy $$ A_+, A_->0,\quad B^2<A_+A_-, \quad t^\pm >0, \quad {t^+}^2+{ t^-}^2=1. $$ If $B<0$, then for every $ε$ small enough, we construct a family of entire solutions $w_ε(\tilde{z}, t)\in \mathbb{C}^2$ in the cylindrical coordinates $(\tilde{z}, t)\in \mathbb{R}^2 \times \mathbb{R}$ for this system via the approach introduced by J. Dávila, M. del Pino, M. Medina and R. Rodiac in {\tt arXiv:1901.02807}. These solutions are $2π$-periodic in $t$ and have multiple interacting vortex helices. The main results are the extensions of the phenomena of interacting helical vortex filaments for the classical (single) Ginzburg-Landau equation in $\mathbb{R}^3$ which has been studied in {\tt arXiv:1901.02807}. Our results negatively answer the Gibbons conjecture \cite{Gibbons conjecture} for the Allen-Cahn equation in Ginzburg-Landau system version, which is an extension of the question originally proposed by H. Brezis.
