  It is known that an explosive instability can occur when nonlinear waves
propagate in certain media that admit 3-wave mixing. The purpose of this paper
is to show that explosive instabilities can occur even in media that admit no
3-wave mixing. Instead, the instability is caused by 4-wave mixing: four
resonantly interacting wavetrains gain energy from a background, and all blow
up in a finite time. Unlike singularities associated with self-focussing, these
singularities can occur with no spatial structure - the waves blow up
everywhere in space, simultaneously.
