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Quantum electrodynamics (QED) effects may be included in physical processesof magnetar and pulsar magnetospheres with strong magnetic fields. Involvingthe quantum corrections, the Maxwell electrodynamics is modified to non-linearelectrodynamics. In this work, we study the force-free magnetosphere innon-linear electrodynamics in a general framework. The pulsar equationdescribing a steady and axisymmetric magnetosphere is derived, which now admitssolutions with corrections. We derive the first-order non-linear corrections tothe near-zone dipole magnetosphere in some popular non-linear effectivetheories. The field lines of the corrected dipole tend to converge on therotational axis so that the fields in the polar region are stronger compared tothe pure dipole case.

The force-free dipole magnetosphere in non-linear electrodynamics