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 {ε}^2 . By employing variational techniques, we establish the existence of solutions for all values of {ε} and get results on their qualitative properties, including regularity. Additionally, we analyse the behaviour of solutions as {ε} {→} 0, demonstrating their pointwise convergence to the solution of the problem for {ε} = 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.