Hyperbolicity, convexity and shock waves in one-dimensional crystalline solids

For a continuum model of one-dimensional anharmonic crystal lattices at finite temperatures, which was derived from a statistical-mechanical model proposed recently, we clarify the classification of its differential system. That is, we determine not only the strict hyperbolicity and convexity regions but also the elliptic and parabolic regions in the space of the state. The melting point is found to be on the boundary of the convexity region. Then we derive the Rankine-Hugoniot relations, and we prove that the admissible shocks are always in the stable region of convexity.