Long time behavior of nonlinear electromagnetic wave in vacuum beyond linear approximation

A quantum nature of vacuum is expected to affect electromagnetic fields in vacuum as a nonlinear correction, yielding nonlinear Maxwell's equations. We extend the finite-difference time-domain (FDTD) method in the case that the nonlinear electromagnetic Lagrangian is quartic with respect to the electric field and magnetic flux density. With this extension, the nonlinear Maxwell's equations can be numerically solved without making any assumptions on the electromagnetic field. We demonstrate examples of self-modulations of nonlinear electromagnetic waves in a one-dimensional cavity, in particular, in a time scale beyond an applicable range of linear approximation. A momentarily small nonlinear correction can accumulate and a comparably large self-modulation can be achieved in a long time scale even though the electromagnetic field is not extremely strong. Further, we analytically approximate the nonlinear electromagnetic waves in the cavity and clarify the characteristics, for example, how an external magnetic flux density changes the self-modulations of phase and polarization.