  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.
