On the problem of smoothness for oscillating solutions of non-strictly hyperbolic systems

Many problems in cold plasma physics are reduced to inhomogeneous quasilinear systems of the non-strictly hyperbolic type, with oscillating solutions. We prove that a necessary condition for global time smoothness of solutions to the Cauchy problem for such systems is the equality of oscillation periods along all characteristics (isochronicity). This property implies, in particular, the non-existence of non-trivial smooth solutions in the problem of plasmaoscillations with a variable doping profile and in the problem of radially symmetric multidimensional plasma oscillations (except for exceptional cases).