The Lopatinski determinant of small shocks may vanish

The Kreiss-Majda Lopatinski determinant encodes a uniform stability property of shock wave solutions to hyperbolic systems of conservation laws in several space variables. This note deals with the Lopatinski determinant for shock waves of sufficiently small amplitude. The determinant is known to be non-zero for so-called extreme shock waves, i. e., shock waves which are asscoiated with either the slowest or the fastest mode the system displays for a given direction of propagation, if the mode is Metivier convex. The result of the note is that for arbitrarily small non-extreme shock waves associated with a Metivier convex mode, the Lopatsinki determinant may vanish.