Abstract A quasilinear system of hyperbolic equations describing plane one-dimensional relativistic oscillations of electrons in a cold plasma is considered. For a simplified formulation, a criterion for the existence of a global-in-time smooth solution is obtained. For the original system, a sufficient condition for singularity formation is found, and a sufficient condition for the smoothness of the solution within the nonrelativistic period of oscillations is established. In addition, it is shown that arbitrarily small perturbations of the trivial solution lead to the formation of singularities in a finite time. The results can be used to construct and substantiate numerical algorithms for modeling the breaking of plasma oscillations.

中文翻译：

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关于双曲问题全局解的存在性

摘要 考虑了描述冷等离子体中电子的平面一维相对论振荡的双曲方程拟线性系统。对于简化的公式，获得全局时间平滑解存在的标准。对于原系统，找到了奇点形成的充分条件，并建立了非相对论振荡周期内解光滑的充分条件。此外，结果表明，平凡解的任意小扰动会导致在有限时间内形成奇点。结果可用于构建和证实用于模拟等离子体振荡破坏的数值算法。