A criterion for the non-existence or non-uniqueness of solutions of self-similar problems in the mechanics of a continuous medium

Abstract

Systems of hyperbolic partial differential equations expressing conservation laws are considered. A sufficient condition is formulated under which the self-similar problem of the disintegration of an arbitrary discontinuity (or the “piston” problem) either has no solution or the solution is not unique. This sufficient condition is determined by the existence of non-evolutionary discontinuities which may be considered as a sequence of two evolutionary discontinuities moving at the same velocity, if such a representation is unique. The condition is more general than that formulated previously, which was based on the existence of a non-proper Jouguet point. The new criterion is satisfied by weak quasitranverse shock waves in elastic media, whatever the sign of the coefficient of the non-linear deformation term. It also enables one to draw conclusions as to the non-existence or non-uniqueness of solutions of problems of the theory of elasticity in the case of finite-amplitude waves.