the Hamilton-Jacobi equation
/ Jacobi -> yah-KOH-bee /
The Hamilton-Jacobi equation is the grand fully-nonlinear first-order PDE that ties together mechanics, optics, and optimal control. In one common form it reads u_t + H(x, grad u) = 0, where u is an unknown function (an action, a travel time, a cost-to-go) and H, called the Hamiltonian, is a given function of position and of the gradient of u. It is nonlinear because H usually depends on grad u in a curved way — squares, square roots, suprema.
It is solved by the method of characteristics in its richest form: characteristic strips, which here are exactly the rays or trajectories of the underlying system. Along each strip the position and the gradient evolve by Hamilton's canonical equations, dx/ds = H_p and dp/ds = -H_x (with p standing for grad u), and the value of u is accumulated along the way. So solving the PDE is literally running classical mechanics along trajectories: the great insight, due to Hamilton and Jacobi, that a hard first-order PDE and a system of ODEs (Hamilton's equations) are two faces of the same theory. In geometric optics the same equation, in the time-independent form H(x, grad u) = 1, is the eikonal equation and its rays are light rays.
Just like Burgers' equation, smooth solutions of Hamilton-Jacobi equations break down: characteristics (rays) cross, the gradient becomes multi-valued, and corners and kinks form (think of where light rays focus, a caustic, or where shortest paths around an obstacle collide). Past that breakdown the right notion of solution is the viscosity solution, the modern framework that restores existence and uniqueness for these fully nonlinear equations, and which underlies optimal control through the Hamilton-Jacobi-Bellman equation.
Shortest travel time across a region with varying speed obeys an eikonal-type Hamilton-Jacobi equation; its characteristic strips are the optimal paths (rays), and where two families of shortest paths meet, the solution u (the arrival time) develops a kink — a wavefront cusp or shock in slope.
Solving an HJ equation runs Hamilton's ODEs along rays; where rays cross, the gradient becomes a kink.
Hamilton-Jacobi solutions are typically only Lipschitz, not smooth — kinks are normal, not pathological. The correct continuation past kinks is the viscosity solution, which (unlike a bare weak solution) is unique; the characteristic-strip method gives the right answer only up to the first ray crossing.