On the criteria of the Langmuir oscillations breaking in a plasma

The influence of the initial distribution of the plasma electron density on the temporal dynamics of nonlinear Langmuir oscillations in plane one-dimensional geometry has been analytically and numerically studied. It is shown that if the concentration of electrons at the initial time in a certain region is less than or equal to half the background value, then the density singularity or breaking always occurs during the first half-period of oscillations in the plane of symmetry with respect to which the electrons oscillate. If the electron concentration at the initial time exceeds half the equilibrium value, then the density singularity appears outside the plane of symmetry due to the relativistic dependence of the oscillation frequency on the amplitude. In this case, the breaking time of oscillations is inversely proportional to the initial value of the electron density perturbations and the square of the electric field at the initial time. (paper)