Quasilinear system

A first-order quasilinear system has the form $U_t+A(U,x,t)U_x=S(U,x,t)$: derivatives of the unknown enter linearly, while their coefficient matrix may depend on the unknown itself.

A quasilinear system is hyperbolic where its principal coefficient matrix has real eigenvalues and enough independent eigenvectors. The eigenvalues are its characteristic speeds.