 Hyperbolicity

For a one-dimensional first-order system $v_t+A(v,x,t)v_x=s$, hyperbolicity means that the coefficient matrix has real eigenvalues and a complete set of eigenvectors. Distinct real eigenvalues give strict hyperbolicity and separate characteristic curves. Along a curve of speed $\lambda$, a left eigenvector $\ell A=\lambda \ell$ gives the compatibility equation $\ell dv/dt=\ell s$.