Hyperbolic partial differential equation

A scalar second-order equationis hyperbolic where $B^2-AC>0$. Equivalently, its principal symbol has two distinct real characteristic directions.

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 $\ell dv/dt=\ell s$.

For a one-dimensional wave equation whose principal symbol is ${\xi }_t^2-c(y{)}^2{\xi }_y^2$, a travel-time coordinate isIts two families of characteristic curves are $s(y)-t=\text{constant}$ and $s(y)+t=\text{constant}$ wherever the wave speed $c$ is positive.

For $c(y)=1-y^2$ in the strip $|y|<1$, the travel-time coordinate is $s(y)=\mathrm{artanh}y$. The characteristic curves areequivalently $y=\tanh (C\mp x)$. Outside the strip the same separated ordinary differential equation gives hyperbolic cotangent branches, and the equilibrium curves $y=\pm 1$ are also characteristic.