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Exact self-similar solutions in Born-Infeld theory

We present a new class of exact self-similar solutions possessing cylindrical or spherical symmetry in Born-Infeld theory. A cylindrically symmetric solution describes the propagation of a cylindrical electromagnetic disturbance in a constant background magnetic field in Born-Infeld electrodynamics. We show that this solution corresponds to vacuum breakdown and the subsequent propagation of an electron-positron avalanche. The proposed method of finding exact analytical solutions can be generalized to the model of a spherically symmetric scalar Born-Infeld field in the ($n+1$)-dimensional Minkowski space-time. As an example, the case $n=3$ is discussed.