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arXiv · 2408.14413

Variational Approach for the Singular Perturbation Domain Wall System

Javier Monreal·Michał Kowalczyk
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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.

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Javier Monreal, Michał Kowalczyk. 2024-08-26. Variational Approach for the Singular Perturbation Domain Wall System. https://arxiv.org/abs/2408.14413

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