The method of characteristics still works when the speed depends on the solution itself. Such equations are quasilinear; the famous example is the inviscid Burgers' equation

a toy model of a fluid where each point moves at a speed equal to its own height. Following the same logic, along curves with \frac{dx}{dt} = u we get \frac{du}{dt} = 0 — so u is still constant along characteristics. The twist: each characteristic now carries its own speed u(x_0, 0), so the lines are no longer parallel.

When characteristics collide

If the initial profile is decreasing somewhere, the taller part behind moves faster than the shorter part ahead. The characteristics converge, and at some finite time they cross. Where two characteristics meet, they try to assign u two different values at one point — impossible. The smooth solution breaks down and a shock forms: a moving jump discontinuity, like the sudden front of a traffic jam or a sonic boom.

This is genuinely new behaviour. Linear equations never do this — their characteristics are parallel and never cross. Nonlinearity is what lets a perfectly smooth start develop a sharp edge on its own.

The lines that crash together

Step through the picture: feet on the x-axis launch characteristics whose slopes are set by the initial height there. Because the left points are taller (faster), the lines lean in and meet — and the first crossing is where the shock is born.