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 \(\varepsilon ^2\) . By employing variational techniques, we establish the existence of solutions for all values of \(\varepsilon \) and get results on their qualitative properties, including regularity. Additionally, we analyse the behaviour of solutions as \(\varepsilon \rightarrow 0\) , demonstrating their pointwise convergence to the solution of the problem for \(\varepsilon =0\) . 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 Painlevé equation.
