Vortex solutions in the Ginzburg-Landau-Painlevé theory of phase transition

Abstract

The extended Painlevé P.D.E. system $Δy -x_1 y - 2 |y|^2y=0$, $(x_1,\ldots,x_n)\in \mathbb{R}^n$, $y:\mathbb{R}^n\to\mathbb{R}^m$, is obtained by multiplying by $-x_1$ the linear term of the Ginzburg-Landau equation $Δη=|η|^2η-η$, $η:\mathbb{R}^{n}\to\mathbb{R}^{m}$. The two dimensional model $n=m=2$ describes in the theory of light-matter interaction in liquid crystals, the orientation of the molecules at the boundary of the illuminated region. On the other hand, the one dimensional model reduces to the second Painlevé O.D.E. $y''-xy-2y^3=0$, $x\in \mathbb{R},$ which has been extensively studied, due to its importance for applications. The solutions of the extended Painlevé P.D.E. share some characteristics both with the Ginzburg-Landau equation and the second Painlevé O.D.E. The scope of this paper is to construct standard vortex solutions $y:\mathbb{R}^{n}\to\mathbb{R}^{n-1}$ ($\forall n\geq 3$) of the extended Painlevé equation. These solutions have in every hyperplane $x_1=\mathrm{Const.}$, a profile similar to the standard vortices $η:\mathbb{R}^{n-1}\to\mathbb{R}^{n-1}$ of the Ginzburg-Landau equation, but their amplitude is determined by the Hastings-McLeod solution $h$ of the second Painlevé O.D.E. evaluated at $x_1$.