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

TL;DR

This paper proves that certain solutions to a generalized Painlevé-II equation minimize an associated energy functional, especially those solutions that are monotonic in one variable and connect specific asymptotic states.

Contribution

It establishes the energy minimality property for solutions of the generalized Painlevé-II equation with specific asymptotic behavior and monotonicity.

Findings

Solutions are energy minimizers among all solutions with similar boundary conditions.

Monotonic solutions converge to Hastings-McLeod solutions at infinity.

Energy minimality characterizes the connecting solutions in the Painlevé 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 $x_2\to \pm \infty$.