a characteristic strip

For a quasilinear equation a characteristic is just a curve — you track where you are and what u is. But for a fully nonlinear first-order equation, where the derivatives appear in complicated ways, knowing the point and the value is not enough to move forward: you also need to carry the slope of the solution at that point. A characteristic strip is a characteristic curve enriched with that extra slope information — a thin ribbon of solution surface, tilted at the right angle, threaded through space.

Write a fully nonlinear first-order equation as F(x, y, u, p, q) = 0, where p = u_x and q = u_y are the partial derivatives. The characteristic strip is carried by the full Charpit-Lagrange system: dx/ds = F_p, dy/ds = F_q (these trace the curve), du/ds = p F_p + q F_q (this lifts it to the surface), and dp/ds = -(F_x + p F_u), dq/ds = -(F_y + q F_u) (these evolve the slope). Five coupled ODEs instead of three. The name strip is apt: at each point of the curve you also carry the little tangent plane (set by p and q), so what propagates is not a bare thread but a thread wearing a tilt — a ribbon. Stitch the strips together and you again recover the solution surface, now of a fully nonlinear equation.

Characteristic strips are how the Hamilton-Jacobi equation of mechanics and geometric optics is solved: the strips are exactly the rays (light rays, particle trajectories), and the slope they carry is the momentum or the wave normal. This is the deep link between PDEs and Hamiltonian mechanics — solving the PDE is running Hamilton's equations along rays — and it generalizes the simple curve-carrying picture of the quasilinear case to the full nonlinear world.