Space-time chaos in spatially continuous systems

We give the natural definition of space-time chaos for extended dynamical systems which are continuous in a (physical) space. Moreover, we show that such a type of dynamics exists in nonlinear wave equations with some broad classes of potentials. This type of space-time chaos is different from the space-time mixing which has been extensively studied for discrete in space extended systems (lattice dynamical systems). It rather describes a chaotic change in time of spatial patterns and corresponds to a topological mixing in a functional space.