Energy minimality property of the connecting solution of the Painlev\'{e} phase transition model

Abstract

We establish the energy minimality property of solutions to the generalized Painlev\'{e}-II equation $\Delta y-x_1y-2y^3=0$, $(x_1,x_2)\in \mathbb{R}^2$, which are increasing in $x_2$ and converge to the positive and negative Hastings-McLeod solutions as $x_2\to \pm \infty$.