Vortex solutions in the Ginzburg-Landau-Painlev\'e theory of phase transition

The extended Painlev'e P.D.E. system $\mathrm{\Delta }y-x_{1}y-2\mathrm{\mid }y{\mathrm{\mid }}^{2}y=0$, $(x_1,\dots ,x_n)\in {\mathbb{R}}^n$, $y:{\mathbb{R}}^n\to {\mathbb{R}}^m$, is obtained by multiplying by $-x_1$ the linear term of the Ginzburg-Landau equation $\mathrm{\Delta }\eta =\mathrm{\mid }\eta {\mathrm{\mid }}^2\eta -\eta$, $\eta :{\mathbb{R}}^n\to {\mathbb{R}}^m$. 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 Painlev'e O.D.E. $y^{\mathrm{\prime }\mathrm{\prime }}-xy-2y^3=0$, $x\in \mathbb{R},$ which has been extensively studied, due to its importance for applications. The solutions of the extended Painlev'e P.D.E. share some characteristics both with the Ginzburg-Landau equation and the second Painlev'e O.D.E. The scope of this paper is to construct standard vortex solutions $y:{\mathbb{R}}^n\to {\mathbb{R}}^{n-1}$ ($\mathrm{\forall }n\ge 3$) of the extended Painlev'e equation. These solutions have in every hyperplane $x_1=\mathrm{C}\mathrm{o}\mathrm{n}\mathrm{s}\mathrm{t}\mathrm{.}$, a profile similar to the standard vortices $\eta :{\mathbb{R}}^{n-1}\to {\mathbb{R}}^{n-1}$ of the Ginzburg-Landau equation, but their amplitude is determined by the Hastings-McLeod solution $h$ of the second Painlev'e O.D.E. evaluated at $x_1$.