Javier Monreal | Searcharxiv

arXiv2025

Yang-Mills instanton on a four dimensional wormhole: asymptotic stability in the energy space

In this paper we consider an $SU(2)$ Yang-Mills field propagating in the $4+1$ dimensional wormhole spacetime. Assuming the spherically symmetric magnetic ansatz the problem reduces to a one dimensional non linear wave equation. This equation posses a degree one solution (instanton) which is odd in space. We consider small, odd perturbations of the instanton and show that it is conditionally asymptotically stable in the odd energy space.

math.AP↗

arXiv2024

Variational Approach for the Singular Perturbation Domain Wall System

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.

math.AP↗