The connecting solution of the Painlevé phase transition model

Abstract

The second Painlevé O.D.E. $y^{\prime \prime }-xy-2y^3=0$, $x\in \mathbb{R},$ is known to play an important role in the theory of integrable systems, random matrices, Bose-Einstein condensates and other problems. The generalized second Painlevé equation $\Delta y-x_{1}y-2y^3=0$, $(x_1,x_2)\in {\mathbb{R}}^2$, is obtained by multiplying by $-x_1$ the linear term $u$ of the Allen-Cahn equation $\Delta u=u^3-u$. It involves a non autonomous potential $H(x_1,y)$ which is bistable for every fixed $x_1<0$, and thus describes as the Allen-Cahn equation a phase transition model. The scope of this paper is to construct a solution $y$ connecting along the vertical direction $x_2$, the two branches of minima of $H$ parametrized by $x_1$. This solution plays a similar role that the heteroclinic orbit for the Allen-Cahn equation. It is the the first to our knowledge solution of the Painlevé P.D.E. both relevant from the applications point of view (liquid crystals), and mathematically interesting.

15 pages, one figure

References in corpus (1)

• Symmetry breaking and restoration in the Ginzburg-Landau model of nematic liquid crystals