Formation of Discontinuities in Classical Nonlinear Electrodynamics

It is shown that discontinuities can develop in the propagation of initially smooth waves governed by a classical nonlinear theory of electrodynamics. The type of theory considered includes as a special case that of Heisenberg and Euler, which describes the modifications that must be made in the Maxwell equations to include the classical limit of the nonlinear vacuum effects of quantum electrodynamics. A particular solution of the equations is constructed by the method of characteristics; this example illustrates how, with the appropriate well-behaved initial conditions, the characteristics can be made to intersect at later times, thus forming discontinuities. The classical approximation fails when the gradient of the field strength becomes large, so that no definite conclusion can be drawn as to the actual physical creation of singularities.