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

考虑了一个双线性方程的拟线性系统的柯西问题，该系统描述了冷等离子体中电子的平面一维相对论振动。对于问题的一些简化表示，获得了存在全局及时解的标准。对于原始问题，找到了一个足够的爆破条件，以及一个至少在时间\（2 \ pi \）上保持溶液光滑的充分条件。。另外，表明在一般情况下，琐碎状态的任意小的扰动会导致在有限时间内形成奇点。进一步证明，存在特殊的初始数据，即使在相对论的情况下，相应的解决方案也始终保持平稳。周期性的行进波给出了这样一种解决方案的例子。为了使这样的波平滑，波的速度必须大于取决于初始数据的某个常数。然而，任意形式的一般形式的细微扰动都会在时间平滑的解决方案中破坏这些全局性。数值例子说明了解奇异性的性质。