Ginzburg-Landau vortex analogues (Q5941967)

Asymptotic behaviour as \(\lambda\to \infty\) of solutions \(u:[-1,1]\to \mathbb{C}/ \{0\}\) of the Dirichlet problem \[ u''=\lambda uV'\bigl(|u|^2 \bigr),\;u(1)=e^{i\Phi},\;u(-1)= e^{-i\Phi}, \] where \(\Phi>0\) is a positive constant and \(V:[0,\infty) \to\mathbb{R}\) a smooth function satisfying \[ V(s)\geq 0\;(0\leq s<\infty),\;V'(s)>0\;(1<s<\infty),\;V(1)=V'(1)=0,\;V''(1)>0 \] is investigated. The result resembles the two-dimensional vortex solutions. In rough terms: the \(\lambda\to\infty\) limit of graphs \(u([-1,1])\) either coincides with the unit circle arc \(\{e^{i\vartheta}, -\Phi<\vartheta <\Phi\}\) or can be obtained from that arc by replacing \(\{e^{i\vartheta},-\pi/2\leq \vartheta\leq \pi/2\}\) with the line segment with endpoints \(\pm i\) (either case is possible if \(\Phi>\pi/2\), only the first subcase is possible if \(0<\Phi <\pi/2)\). Even more interesting results are obtained for the equation \(u''=\lambda u(|u |^2-1)\) on the circle \(S^1\), where \(u:S^1\to \mathbb{C}/ \{0\}\) is assumed to be of a fixed degree \(N\).