  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.
