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We use the well-known Bogomolny's equations, in general coordinate system,for BPS monopoles and dyons in the $SU(2)$ Yang-Mills-Higgs model to obtain anexplicit form of BPS Lagrangian density under the BPS Lagrangian method. Wethen generalize this BPS Lagrangian density and use it to derive severalpossible generalized Bogomolny's equations, with(out) additional constraintequations, for BPS monopoles and dyons in the generalized $SU(2)$Yang-Mills-Higgs model. We also compute the stress-energy-momentum tensor ofthe generalized model, and argue that the BPS monopole and dyon solutions arestable if all components of the stress-tensor density are zero in the BPSlimit. This stability requirement implies the scalar fields-dependent couplingsto be related to each other by an equation, which is different from the oneobtained in~\cite{Atmaja:2018cod}, and then picks particular generalizedBogomolny's equations, with no additional constraint equation, out of thosepossible equations. We show that the computations in~\cite{Atmaja:2018cod} areactually incomplete. Under the Julia-Zee ansatz, the generalized Bogomolny'sequations imply all scalar fields-dependent couplings must be constants, whosesolutions are the BPS dyons of the $SU(2)$ Yang-Mills-Higgsmodel~\cite{Prasad:1975kr}, or in another words there are no generalized BPSdyon solutions under the Julia-Zee ansatz. We propose two possible ways forobtaining generalized BPS dyons, where at least one of the scalarfields-dependent couplings is not constant, that are by using differentansatze, such as axially symmetric ansatz for higher topological charge dyons;and/or by considering the most general BPS Lagrangian density.

Are There BPS Dyons in The Generalized $SU(2)$ Yang-Mills-Higgs Model?