characteristic coordinates
Sometimes a problem looks hopeless in the coordinates you were handed and becomes trivial in coordinates chosen to fit it — describe a tilted ellipse in its own axes and the equation snaps clean. Characteristic coordinates are precisely such a tailored coordinate system for a second-order PDE: new variables laid down ALONG the characteristic curves, in which the equation reaches its canonical form. They are the practical machinery behind reducing an equation to its type's model.
Concretely, suppose the characteristic ODE of a hyperbolic equation has two families, phi(x,y) = const and psi(x,y) = const. You define the new variables xi = phi(x,y) and eta = psi(x,y) — each constant exactly along one characteristic family. Rewriting all the derivatives by the chain rule (u_x and u_y become combinations of u_xi and u_eta, and similarly the second derivatives) and substituting back, the algebra is arranged so that the u_xi xi and u_eta eta terms cancel, leaving only the mixed term: the equation becomes u_{xi eta} + (lower order) = 0. For a parabolic equation the single characteristic supplies one coordinate and you pick any smooth second coordinate independent of it; for an elliptic equation the characteristics are complex conjugates, so you take xi and eta to be their real and imaginary parts.
These coordinates are not a mere trick — they reveal the equation's true grain. In characteristic coordinates a hyperbolic solution visibly splits into pieces that depend on xi alone and on eta alone (the left- and right-movers), the domain of dependence becomes a simple coordinate rectangle, and integration along the characteristics is straightforward. The honest caveats: with variable coefficients the characteristics curve and the new coordinates are valid only locally (and only where the change of variables is invertible, i.e. its Jacobian does not vanish), and for an equation that changes type the construction breaks down where the type changes.
For u_tt - c^2 u_xx = 0, set xi = x - ct and eta = x + ct. The chain rule gives u_tt = c^2(u_xi xi - 2u_xi eta + u_eta eta) and c^2 u_xx = c^2(u_xi xi + 2u_xi eta + u_eta eta); subtracting leaves -4c^2 u_xi eta = 0, i.e. u_xi eta = 0. Two everyday variables (x and t) traded for two characteristic ones (xi and eta), and the wave equation becomes child's play.
Characteristic coordinates are the explicit change of variables that carries an equation to its canonical form.
Characteristic coordinates exist as real variables only when real characteristics do — that is, for hyperbolic and (one of them) parabolic equations; for an elliptic equation the natural change of variables is complex, which is why one passes to the real and imaginary parts.