Abstract
We study 2×2 systems of hyperbolic conservation laws near an umbilic point. These systems have Undercompressive shock wave solutions, i.e., solutions whose viscous profiles are represented by saddle connections in an associated family of planar vector fields. Previous studies near umbilic points have assumed that the flux function is a quadratic polynomial, in which case saddle connections lie on invariant lines. We drop this assumption and study saddle connections using Golubitsky-Schaeffer equilibrium bifurcation theory and the Melnikov integral, which detects the breaking of heteroclinic orbits. The resulting information is used to construct solutions of Riemann problems.
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Schecter, S., Shearer, M. Undercompressive shocks for nonstrictly hyperbolic conservation laws. J Dyn Diff Equat 3, 199–271 (1991). https://doi.org/10.1007/BF01047709
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DOI: https://doi.org/10.1007/BF01047709