the characteristic ODE system

The method of characteristics promises to replace one partial differential equation with a handful of ordinary ones. The characteristic ODE system is that handful written out: the precise set of coupled ODEs that say how the position and the solution value move together as you travel along a characteristic curve, parametrized by an auxiliary variable usually called s.

For the quasilinear equation a(x,y,u) u_x + b(x,y,u) u_y = c(x,y,u), the system is dx/ds = a, dy/ds = b, du/ds = c. Read it as a recipe for building the solution surface point by point: start at a point on the initial curve, step forward by ds, and these three equations tell you simultaneously how x moves, how y moves, and how the solution value u changes. The first two equations trace the characteristic curve in the plane; the third lifts it to the right height, drawing the solution surface as the characteristic sweeps along. Crucially, because a, b, c can depend on u, all three equations are coupled and must be solved together — the geometry and the solution value are intertwined.

This system is the operational heart of first-order PDEs. It is what you actually integrate. Different first-order problems differ only in the functions a, b, c (and, for fully nonlinear equations, in a longer system that also tracks the gradient — the characteristic strip equations). When the system can be integrated in closed form you get an exact solution; when it cannot, it still tells you the geometry of propagation and is the basis for numerical schemes that march along characteristics.