Parametric excitation of high-mode oscillations for a non-linear telegraph equation

The problem of parametric excitation of high-mode oscillations is solved for a non-linear telegraph equation with a parametric external excitation and small diffusion. The equation is considered on a finite (spatial) interval with Neumann boundary conditions. It is shown that under a proper choice of parameters of the external excitation this boundary-value problem can have arbitrarily many exponentially stable solutions that are periodic in time and rapidly oscillate with respect to the spatial variable.