The connecting solution of the Painlev\'e phase transition model

The second Painlev\'e O.D.E. $y''-xy-2y^3=0$, $x\in \mathbb{R},$ is known to play an important role in the theory of integrable systems, random matrices, Bose-Einstein condensates and other problems. The generalized second Painlev\'e equation $\Delta y -x_1 y - 2 y^3=0$, $(x_1,x_2)\in \mathbb{R}^2$, 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 Painlev\'e P.D.E. both relevant from the applications point of view (liquid crystals), and mathematically interesting.