One small relaxation of the rules
In the last guide you met the method of characteristics on its friendliest home ground: a linear first-order PDE like a(x,y) u_x + b(x,y) u_y = c(x,y), where the coefficients a and b are functions of position only. Because they never look at the unknown u, you could draw the whole web of characteristic curves first, then pour the data along them. The roads were paved before any traffic arrived.
A quasilinear equation makes exactly one change, and it is deceptively small. The first derivatives u_x and u_y still appear linearly — no squares of u_x, no product u_x u_y — but their coefficients are now allowed to depend on u itself. The general two-variable form is a(x,y,u) u_x + b(x,y,u) u_y = c(x,y,u). Compare it side by side with the linear case and the only difference is that extra u sitting inside a and b.
It helps to see three rungs of the same ladder side by side. A linear equation has coefficients a(x,y) and b(x,y) that ignore u entirely. A semilinear equation lets only the right-hand side see u — a(x,y) u_x + b(x,y) u_y = c(x,y,u) — so the characteristics are still fixed in advance. The quasilinear equation goes one step further and lets a and b themselves depend on u. As you read from linear to semilinear to quasilinear, u creeps deeper into the equation, and the characteristics lose their independence from the solution.
That one extra u changes everything downstream. The equation can now feed back on its own solution: how fast information travels depends on the value being carried. This is the doorway to genuinely nonlinear behaviour — steepening waves, crossing characteristics, and eventually shocks — and it is why this guide exists between the gentle linear story and the dramatic one to come.
The characteristic ODEs, now coupled
The mechanism is the same as before, with one twist. We parametrise a characteristic curve by an auxiliary variable s and ask: as we step along it, how do x, y, and the solution value u all move together? Matching the PDE term by term gives the characteristic ODE system: dx/ds = a, dy/ds = b, du/ds = c. The first two equations trace the curve in the plane; the third lifts it to the right height.
dx/ds = a(x,y,u) dy/ds = b(x,y,u) <-- a and b now contain u du/ds = c(x,y,u)
Here is the crucial difference. In the linear case dx/ds and dy/ds knew nothing about u, so you could integrate the first two equations alone, get the curves, and only afterwards carry u along. Now a and b contain u, so all three equations are tangled together — you cannot find the curve without knowing the value, and you cannot find the value without knowing the curve. They must be solved simultaneously, as one coupled system.
Why a surface, not just curves
The cleanest way to see what is happening is to stop thinking about formulas and start thinking about a shape. A solution u = u(x,y) is the same thing as a surface floating above the (x,y)-plane: at each spot, the height of the surface is the solution value. Building the solution means weaving this surface into existence, and the characteristics are the threads you weave it from.
The geometry is exact and worth holding onto. Rewrite a u_x + b u_y = c as a u_x + b u_y - c = 0. The surface u = u(x,y) has normal vector (u_x, u_y, -1), and this equation says the vector (a, b, c) is perpendicular to that normal — that is, (a, b, c) lies tangent to the surface at every point. But (a, b, c) is exactly the tangent direction of a characteristic, from the ODEs dx/ds = a, dy/ds = b, du/ds = c. So every characteristic curve lies entirely within the solution surface. They are stitched into it.
This picture makes the whole method honest. To build the surface you take an initial data curve — one thread already drawn in space — and through each of its points you run a characteristic. The family of characteristics sweeps out the full surface, the way a comb's teeth sweep out a sheet. One thread alone could never fill a surface, which is exactly why prescribing data along a single characteristic fails: it hands you one thread and asks for a sheet.
A worked example you can carry by hand
Take the simplest genuinely quasilinear first-order equation there is: u_t + u u_x = 0, the inviscid form of Burgers' equation. Read it as a transport equation whose carrying speed is u itself — tall parts of the profile move faster than short parts. Let us run the characteristics and watch the consequence appear.
- Write the characteristic ODEs. With (x, t) as the plane and the equation u_t + u u_x = 0, the system is dt/ds = 1, dx/ds = u, du/ds = 0. (We took the t-coefficient as 1.)
- Read du/ds = 0: the value u stays constant along each characteristic. So whatever value u starts with at t = 0, it keeps forever along its own curve.
- Because u is constant on the curve, dx/ds = u is a fixed number, so the characteristic is a straight line: x = x0 + u(x0,0) t. Its slope is set by the initial value at its foot.
- Now picture two starting points: one where the data is large, one just ahead where the data is smaller. The large-value line is steeper-fast and the small-value line is slow — so the fast line catches the slow line. The characteristics cross.
That is the entire plot of nonlinear first-order PDEs in miniature. Up to the first crossing time the smooth solution is perfectly fine. At the crossing, two characteristics deliver two different values of u to the same point (x, t) — and a single-valued solution cannot honour both. The surface from the previous section would have to fold over itself, taking two heights above one spot, which no graph can do.
What this rung is really teaching you
Step back and notice the shift in worldview. For a linear equation the characteristic curves are a fixed map drawn by the coefficients, and the solution is just cargo carried along fixed roads. For a quasilinear equation the cargo and the roads are the same thing: the value you carry decides where the road goes. That feedback is the whole source of nonlinear surprise.
This also reframes a habit from earlier rungs. With ODEs, the general solution carried arbitrary constants and data fixed the constants. With a PDE the general solution carries arbitrary functions, and data — here, values prescribed along a curve — is what pins down which function. The next guide makes that precise: not every data curve will do, and the rule that separates the good curves from the bad is the non-characteristic condition.