Exact BPS domain walls at finite gauge coupling

Bogomol'nyi-Prasad-Sommerfield solitons in models with spontaneously broken gauge symmetry have been intensively studied at infinite gauge coupling limit, where the governing equation -- so-called master equation -- is exactly solvable. Except for a handful of special solutions, the standing impression is that analytic results at finite coupling are generally unavailable. The aim of this paper is to demonstrate, using domain walls in Abelian-Higgs models as the simplest example, that exact solitons at finite gauge coupling can be readily obtained if the number of Higgs fields ($N_F$) is large enough. In particular, we present a family of exact solutions, describing $N$ domain walls at arbitrary positions in models with at least $N_F \geq 2N+1$. We have also found that adding together any pair of solution can produce a new exact solution if the combined tension is below a certain limit.