First-Order PDEs and Characteristics

The transport equation is simply the rule for something being carried along without changing, like dye drifting down a river that flows at speed $c$, with no spreading at all. It's called "transport" because that's all it does: it moves the profile from place to place. It's good for anything that gets swept along, like pollutants in a current, cars on a highway or signals on a wire, and it's the easiest PDE to solve completely. The equation $u_t + c\,u_x = 0$ has just two pieces: $u_t$ (how fast the color changes if you stand still on the bank) and $c\,u_x$ (how fast it changes because the current is sweeping new water past you). If you float along with the current, the color around you never changes, so the two cancel out, and that's the whole equation.

Why it works. $u_t + c\,u_x$ is the derivative of $u$ along the direction $(1, c)$ in the $(t, x)$-plane. You can see it with the chain rule: if you move so that $x = x_0 + ct$, then $\frac{d}{dt}u(x_0 + ct, t) = u_t + c\,u_x$. The PDE says that's zero, so $u$ doesn't change as you move along the lines $x = x_0 + ct$. Each such line carries the initial value $f(x_0)$ forward, and since $x_0 = x - ct$, the solution is $u = f(x - ct)$. There's a graph further down where you can drag $t$ and watch a bump slide.

A common mistake. It's tempting to read $f(x - ct)$ as moving left, because of the minus sign. It isn't. The peak of $f$ sits wherever $x - ct = 0$, which is at $x = ct$, and that moves right when $c > 0$. You have to subtract to "undo" the motion and look back at where the value started. A negative $c$, as in $u_t - 2u_x = 0$, is what sends the profile left.

Variable speed. Now suppose the current is faster in some places than others. The lines bend into curves: for $u_t + a(x, t)\,u_x = 0$, solve $\frac{dx}{dt} = a$ to get the characteristic curves, and $u$ is constant on each. Notice that finding them is just a first-order ODE. For $u_t + x\,u_x = 0$, the curves are $x = x_0e^t$, so $u = f(xe^{-t})$.

Nonlinear equations. In Burgers' equation $u_t + u\,u_x = 0$ the speed is $u$ itself, so tall parts of a wave move faster than low parts. The tall parts catch up with the low parts ahead of them, characteristics cross, and since $u$ can't take two values at one point, a shock forms. This models traffic jams and breaking waves.

Pollutants carried by a river, traffic density on a highway, and signals on transmission lines move by transport equations; shocks in Burgers' equation model traffic jams and sonic booms.