Burgers' equation
/ Burgers -> BUR-gers /
Burgers' equation is the simplest equation that shows a smooth wave steepening, breaking, and forming a shock — and for that reason it is the laboratory animal of nonlinear PDEs. It is a single scalar equation, almost as plain as the transport equation, yet it already contains the whole drama of conservation laws: characteristics crossing, gradient blow-up, shocks, rarefactions, and the entropy condition.
The inviscid form is u_t + u u_x = 0. It is the transport equation with one telling change: the speed at which u is carried is u itself. So a tall part of the profile races ahead while a short part lags, and the wave leans forward and steepens. Written as a conservation law it is u_t + (u^2/2)_x = 0, so the flux is u^2/2. Its characteristics are the straight lines x = x0 + u(x0,0) t — slope set by the data — which cross when the data decreases, producing a shock at a finite time. The viscous form adds a small smoothing term, u_t + u u_x = nu u_xx, which keeps solutions smooth for all time and resolves the shock into a thin steep layer; letting nu shrink to zero (vanishing viscosity) recovers the inviscid shock and singles out the physically correct, entropy-satisfying weak solution.
Burgers' equation matters far beyond its own modest physics. It is the canonical test problem for shock theory, for the Rankine-Hugoniot jump condition that fixes a shock's speed, for the entropy condition that restores uniqueness, and for numerical schemes (which must capture shocks without spurious wiggles). Master Burgers and you have the skeleton of the entire theory of scalar conservation laws.
Start the inviscid Burgers equation with a step that goes from u = 1 on the left down to u = 0 on the right. The faster left state overtakes the right, so the characteristics collide immediately and a shock forms at once, moving at the average speed (1 + 0)/2 = 1/2 — exactly the Rankine-Hugoniot speed.
A down-step gives an instant shock at speed 1/2; an up-step instead opens a rarefaction fan.
For the same up-step data both a moving shock and a spreading rarefaction are weak solutions, so a weak solution is not unique. The entropy condition rejects the (unphysical, characteristic-emitting) shock and keeps the rarefaction; without it, uniqueness is lost.