The Rankine–Hugoniot conditions and balance laws for $\delta$-shocks

New definitions of $\delta$-shock wave-type solutions are introduced for two (one-dimensional) types of hyperbolic systems of conservation laws. Corresponding Rankine–Hugoniot conditions for $\delta$-shocks are derived, and their geometrical interpretation is given. Balance laws connected with “area,” mass, and momentum transportation for $\delta$-shocks are derived.