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 Melnikov's 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. (C) 2000 Academic Press.