We derive the equations of motion governing static dyonic matters, described
in terms of two real scalar fields, in nonlinear electrodynamics of the
Born–Infeld theory type. We then obtain some exact finite-energy solutions of
these equations in a few interesting special situations subject to dyonic
point-charge sources and construct dyonically charged black holes with
relegated curvature singularities. In the case of quadratic nonlinearity, in
particular, we show that dyonic solutions enable us to restore electromagnetic
symmetry, which is known to be broken in non-dyonic situations. We further
demonstrate that in the context of k-essence cosmology the nonlinear
electrodynamics models possess their own distinctive signatures in light of the
underlying equations of state of the cosmic fluids they represent. In
particular and more importantly, the generalized models here are shown to
resolve a density-pressure inconsistency issue exhibited by the original
Born–Infeld model k-essence action function as well as by all of its
fractional-powered extensions.

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