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A quantum nature of vacuum is expected to affect electromagnetic fields invacuum as a nonlinear correction, yielding nonlinear Maxwell's equations. Weextend the finite-difference time-domain (FDTD) method in the case that thenonlinear electromagnetic Lagrangian is quartic with respect to the electricfield and magnetic flux density. With this extension, the nonlinear Maxwell'sequations can be numerically solved without making any assumptions on theelectromagnetic field. We demonstrate examples of self-modulations of nonlinearelectromagnetic waves in a one-dimensional cavity, in particular, in a timescale beyond an applicable range of linear approximation. A momentarily smallnonlinear correction can accumulate and a comparably large self-modulation canbe achieved in a long time scale even though the electromagnetic field is notextremely strong. Further, we analytically approximate the nonlinearelectromagnetic waves in the cavity and clarify the characteristics, forexample, how an external magnetic flux density changes the self-modulations ofphase and polarization.

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