Some exact BPS solutions for exotic vortices and monopoles

We present several analytical solutions of BPS vortices and monopoles in the generalized Abelian Maxwell–Higgs and Yang–Mills–Higgs theories, respectively. These models have recently been extensively studied and several exact solutions have already been obtained in [1] and [2]. In each theory, the dynamics is controlled by the additional two positive scalar-field-dependent functions, f(|ϕ|)f(|ϕ|) and w(|ϕ|)w(|ϕ|). For the case of vortices, we work in the ordinary symmetry-breaking Higgs potential, while for the case of monopoles we have the ordinary condition of the Prasad–Sommerfield limit. Our results generalize the exact solutions found previously. We also present solutions for BPS vortices with higher winding number. These solutions suffer from the condition that w(|ϕ|)w(|ϕ|) has negative value at some finite range of r, but we argue that since it satisfies the weaker positive-value conditions then the corresponding energy density is still positive-definite and, thus, they are acceptable BPS solutions.