Solutions in Nonlinear Electrodynamics and their double copy regular black holes

We study solutions in non-linear electrodynamics (NED) and establish several general results. We show, that the SO(2) electric-magnetic duality symmetry is restrictive enough to allow for reconstruction of the NED Lagrangian from the spherically-symmetric electrostatic (Coulomb-like) solution - although there are infinitely many different NED theories admitting a given solution, there exists a unique SO(2) invariant one among them under a simple analyticity assumption (that leaves out some interesting physical theories). We introduce a general algorithm for constructing new SO(2) invariant NED theories in the conventional approach, where only a few examples are available. We also show how to derive the Lagrangian of the SO(2) invariant theory admitting a given electrostatic solution. We further show on a simple example that some NED theories may require sources (particles) of finite (non-zero) size. Such a non-zero size source not only regularizes the infinite energy of the point charge but also satisfies the condition of regularity, that the electric field is zero at the origin. The latter condition was identified earlier as necessary and sufficient for the NED solution to generate a regular black hole via so-called double copy construction and is also satisfied by solitons. We propose a large class of solitonic NED solutions that give rise to regular black holes via double copy construction and contain solutions of Maxwell and Born-Infeld as different limits. This class of NED solutions acquires two new properties in the limit where the corresponding regular black hole's asymptotics becomes Minkowski: it gives rise to regular higher-spin black holes via generalization of double copy - "higher-copy" construction, and for very short distances changes the sign of the force becoming repulsive/attractive for opposite/similar signs of charges.