On the conditions for the breaking of oscillations in a cold plasma

The Cauchy problem for a quasilinear system of hyperbolic equations describing plane one-dimensional relativistic oscillations electrons in cold plasma is considered. For some simplified formulation the problem, criterion existence global time solutions obtained. original sufficient condition loss smoothness found, as well solution to remain smooth at least $ 2 \pi $. In addition, it shown that general case, arbitrarily small perturbations trivial state lead formation singularities finite time. It further proved there are special initial data such respective remains all time, even case. Periodic space traveling wave gives an example solution. order be smooth, velocity must greater than certain constant depends on data. Nevertheless, arbitrary perturbation form destroys these solutions. nature illustrated by numerical examples.