Generalized Rankine–Hugoniot Condition and Shock Solutions for Quasilinear Hyperbolic Systems

For a quasilinear hyperbolic system, we use the method of vanishing viscosity to construct shock solutions. The solution consists of two regular regions separated by a free boundary (shock). We use Melnikovʹs integral to obtain a system of differential/algebraic equations that governs the motion of the shock. For Lax shocks in conservation laws, these equations are equivalent to the Rankine–Hugoniot condition. For under compressive shocks in conservation laws, or shocks in non-conservation systems, the Melnikov-type integral obtained in this paper generalizes the Rankine–Hugoniot condition. Under some generic conditions, we show that the initial value problem of shock solutions can be solved as a free boundary problem by the method of characteristics