Nonlinear $\arcsin$-electrodynamics and asymptotic Reissner-Nordstr\"om black holes

A model of nonlinear electrodynamics with the Lagrangian density ${\cal L} = -(1/\beta)\arcsin(\beta F_{\mu\nu}F^{\mu\nu}/4)$ is proposed. The scale invariance and the dual invariance of electromagnetic fields are broken in the model. In the limit $\beta\rightarrow 0$ one comes to Maxwell's electrodynamics and the scale and dual invariances are recovered. We investigate the effect of coupling electromagnetic fields with the gravitational field. The asymptotic black hole solution is found which is similar to the Reissner-Nordstr\"om solution. We obtain corrections to Coulomb's law and to the Reissner-Nordstr\"om solution in the model proposed. The existence of the regular asymptotic at $r\rightarrow 0$ was demonstrated. The mass of the black hole is calculated possessing the electromagnetic origin. It was shown that there are not superluminal fluctuations and principles of causality and unitarity take place.