Reduction procedure and generalized simple waves for systems written in Riemann variables
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.
scientific article; zbMATH DE number 1892292