Formation of Singularities in One-Dimensional Nonlinear Wave Propagation

The waves considered here are solutions of a first-order strictly hyperbolic system of differential equations, written in the form (1) $${u_t} + a(u){u_x} = 0$$ , where u = u(x, t) is a vector with n components u 1,· ·· , u n depending on two scalar independent variables x, t, and a = a(u) is an n-th order square matrix. The question to be discussed is the formation of singularities of a solution u of (1) corresponding to initial data (2) u(x, 0) = f(x). It will be shown that if the system (1) is "genuinely nonlinear" in a sense to be defined below, and if the initial data are "sufficiently small" (but not identically 0), the first derivatives of u will become infinite for certain (x, t) with t > 0. The result is well known for n = 1 and n = 2 (see the papers by Lax and by Glimm and Lax [1], [2], [10]). In many cases such a singular behavior can be identified physically with the formation of a shock, and considerable interest attaches to the study of the subsequent behavior of the solution. In the present paper we shall be content just to reach the onset of singular behavior, without attempting to define and to follow a solution for all times. For that we would need a physical interpretation of our system to guide us in formulating proper shock conditions.