Reduction procedure and generalized simple waves for systems written in Riemann variables (Q1866123)

The authors of this interesting paper study quasilinear hyperbolic systems of conservation laws, which have the form, \(\partial u/\partial t+Q\partial u/\partial x =f\), where \(Q=Q(x,t,u)\) is an \(m\times m \) matrix, \(f=f(x,t,u)\) is a vector-function, \(x\in \mathbb{R}\) and \(t\in\mathbb{R}_+\) are real independents, and \(u\) is some unknown solution. There are first-order differential constraints \(\Phi_k(x,t,u,u_x)=0\) \((k=1,2,\dots ,q)\) which satisfy the natural requirement \(\text{rank}(\partial \Phi_k/\partial u_x)=q\). A solution of the system under consideration, which satisfies the above stated constraints, is called a solution characterized by differential constraints. The system with constraints is an overdetermined system and the known method of solving requires some compatibility. A class of solutions called generalized simple waves is investigated. The main feature of these solutions is that they are constructed by integrating a system of ordinary differential equations. The solutions of a system of two equations reduced in Riemann variables can be obtained by integrating some simple equations. The mathematical models of traffic flow and rate-type materials are analyzed as examples by which the above method is illustrated. Exact solutions called generalized simple waves are obtained.