strict hyperbolicity

If two waves travel at exactly the same speed they can be hard to tell apart and hard to disentangle. Strict hyperbolicity is the clean, friendly case where every wave moves at its own distinct speed, with no ties. It is the hypothesis under which the theory of conservation laws works most smoothly, because each wave family keeps its own lane.

Precisely, a system U_t + A(U) U_x = 0 is strictly hyperbolic at a state U if the matrix A(U) has n real eigenvalues that are all distinct: lambda_1(U) < lambda_2(U) < ... < lambda_n(U). Distinct eigenvalues automatically give a full set of independent eigenvectors, so strict hyperbolicity implies hyperbolicity — but it is a stronger, cleaner condition. With strict separation the eigenvalues and eigenvectors depend smoothly on U, the n characteristic fields are well-defined everywhere, and the elementary waves (shocks, rarefactions, contact discontinuities) of each family can be analysed one family at a time. This is the setting of Lax's theorem on the local solvability of the Riemann problem for small jumps.

Where it matters: strict hyperbolicity is the standing assumption behind most of the classical existence and uniqueness theory for systems of conservation laws (for instance Glimm's theorem on global solutions of small total variation). The honest caveat is that important real systems are not strictly hyperbolic everywhere: in gas dynamics the eigenvalues can coincide, and in multi-dimensional or resonant problems eigenvalues cross. Where they coincide the eigenvectors can degenerate and the neat one-family-at-a-time picture breaks down, which is exactly where the hard open problems live.