Variational Approach for the Singular Perturbation Domain Wall System

TL;DR

This paper investigates a coupled differential system with singular perturbation, establishing existence, regularity, and convergence of solutions as the perturbation parameter approaches zero, and relates it to the second Painlevé equation.

Contribution

It introduces a variational method to prove existence and analyze the asymptotic behavior of solutions in a singular perturbation domain wall system.

Findings

Solutions exist for all perturbation parameters.

Solutions converge pointwise as perturbation vanishes.

Relation to the second Painlevé equation is established.

Abstract

In this article we study a coupled system of differential equations with Allen-Cahn type non-linearity. Motivated by physical phenomena one of the unknowns in the system is accompanied by a singular perturbation parameter ${\epsilon}^2$ . By employing variational techniques, we establish the existence of solutions for all values of ${\epsilon}$ and get results on their qualitative properties, including regularity. Additionally, we analyse the behaviour of solutions as ${\epsilon} {\to} 0$, demonstrating their pointwise convergence to the solution of the problem for ${\epsilon} = 0$. We establish the uniqueness of this solution modulo translations. Additionally, in the final section, through an appropriate change of scale, we relate this problem and the second Painlev\'e equation.