The simplest model of cold plasma

We consider the reduction of the system of Euler-Maxwell equations describing the so-called cold (electron) plasma to a one-dimensional case. This is the simplest system of equations capable of describing plasma oscillations and their breaking. It is interesting because it allows for analytical consideration. Physicists are primarily interested in the conditions on the initial data under which the solution to the Cauchy problem can exist for as long as possible. However, the one-dimensional system of equations for cold plasma is also very interesting mathematically: it is inhomogeneous, not strictly hyperbolic, and is not written in a conservative form. We will discuss issues related to finding a criterion for the formation of singularities in terms of initial data, factors that can extend the life of a smooth solution, and the possibility of the existence of traveling waves. In addition, we will discuss the construction of a solution to the Riemann problem, which in this case is very specific: rarefaction waves and shock waves replace each other, the rarefaction wave is constructed in a non-unique way, and to construct a shock wave one has to use the technique of constructing strongly singular generalized solutions.