Embedded delta shocks

TL;DR

This paper explores new types of delta shocks in conservation laws, including solutions with central rarefactions and without intermediate states, expanding the understanding of singular shock solutions.

Contribution

It introduces a broader class of two-family wave solutions that include delta shocks and central rarefactions, extending previous models of shock discontinuities.

Findings

Identified new solution structures with delta shocks and rarefactions

Extended the theory of delta shocks to more complex wave interactions

Provided a framework for analyzing non-constant intermediate states

Abstract

In 1977 Korchinski presented a new type of shock discontinuity in conservation laws. These singular solutions were coined $\delta$-shocks since there is a time dependent Dirac delta involved. A naive description is that such $\delta$-shock is of the overcompressive type: a two-family shock wave the four characteristic lines of which impinge into the shock itself. In this work, we open the fan of solutions by studying two-family waves without intermediate constant states but, possessing central rarefactions and also comprising $\delta$-shocks.