7.6.1. The Method of Characteristics

In this section we consider hyperbolic systems

in the strip 0 ? x ?1, t ?0 with initial data

At each boundary point x=0, x=1 we prescribe (inhomogeneous) linear relations for u; i.e., we consider boundary conditions of the general form

Specific assumptions about the matrices L ₀, L ₁ will be derived below. The coefficients and data functions are assumed to be C ^?-smooth with respect to all variables.

Hyperbolicity of the system (7.6.1) requires the eigenvalues ? _j (x, t) of B(x, t) to be real and assumes the existence of a smooth transformation S= S(x, t) such that

In the interior 0< x<1 the eigenvalues may change sign. However, we assume that

have a constant sign as a function of time; i.e., each function (7.6.4) is either >0 for all t,=0 for all t, or <0 for all t.

If we introduce new variables (so-called characteristic variables)

then the system (7.6.1) transforms to an equation where B is replaced by ?. To simplify notation, we assume that the given system is written in characteristic variables already, thus B= ? in (7.6.1).

The case of n scalar equations. To discuss the system, we use the method of characteristics and start with the case C= F=0. The differential system separates into n