On the stability of one-dimensional stationary solutions of hyperbolic systems of differential equations containing points at which one characteristic velocity becomes zero

Abstract

The stability of the stationary solutions of hyperbolic systems of partial differential equations containing a point at which one of the characteristic velocities becomes zero, is investigated. The functions sought are assumed to be time and coordinate dependent, and their number is arbitrary. The study of stability carried out below is based on the results obtained in /1, 2/, according to which the behaviour of the unsteady perturbations near the critical point is described by a single non-linear partial differential equation irrespective of the number of equations in the initial system. The equation is written in terms of a function analogous to the Riemann invariant connected with the vanishing characteristic velocity. The equation is used below to examine all possible cases of continuous solutions of an arbitrary hyperbolic system of equations with continuous and discontinuous right-hand sides, and conditions are formulated under which the growth of perturbations near the critical point at which one of the characteristic velocities becomes zero, leads to the instability of the whole solution in toto. The investigation is carried out taking into account the onset and development of the perturbations connected with other characteristic velocities which have a constant sign within the region considered.