Characteristics and Canonical Forms

From a label to a normal form

In the previous guide the discriminant B^2 - A C handed you a single word for any second-order equation A u_xx + 2 B u_xy + C u_yy + (lower-order terms) = 0: elliptic when B^2 - A C < 0, parabolic when it equals 0, hyperbolic when it is positive. That word is a true invariant — it survives any smooth change of coordinates. But knowing the label and seeing it are different things. This guide turns the label into something you can read straight off the page by choosing the right coordinates.

The idea is the same one you trust from linear algebra. A messy quadratic form can be rotated and stretched until its matrix becomes diagonal with neat plus and minus ones on the diagonal — its signature. The principal part A u_xx + 2 B u_xy + C u_yy is exactly such a quadratic form in the derivative directions, and its signature is precisely what the discriminant measures. So the plan is to find a change of variables that diagonalises it, leaving behind one of three clean templates we call canonical forms.

Characteristics: the curves that refuse to be normal

The right coordinates are not random — they are dictated by the equation itself, through its characteristic curves. You met characteristics in the first-order rung as the roads along which information travelled. For a second-order equation they reappear with a sharper job: they are the curves along which the second derivative is not pinned down by the data, the directions in which the equation refuses to behave like an ordinary one. Bending your axes onto these special curves is what makes the principal part collapse.

There is one tidy equation that finds them. A curve given by phi(x,y) = constant is characteristic exactly when its slope dy/dx satisfies the characteristic equation A (dy/dx)^2 - 2 B (dy/dx) + C = 0. This is just a quadratic in the slope, and the quantity under its square root is — no surprise at all — the discriminant B^2 - A C. The three types are simply the three things a quadratic can do: two real roots, one repeated root, or no real roots.

Count the real roots and you have read off the type. When B^2 - A C > 0 the quadratic has two real slopes, so there are two characteristic families and the equation is hyperbolic. When B^2 - A C = 0 the two slopes coincide into one repeated root — a single family — and the equation is parabolic. When B^2 - A C < 0 there are no real slopes at all, no real characteristics, and the equation is elliptic. The arithmetic of a high-school quadratic and the deepest split in the subject are the same calculation.

This is the geometric heart of the whole trichotomy. A hyperbolic equation has two distinct families of characteristics criss-crossing the plane — the natural grid for the wave equation. A parabolic equation has the two families fused into one, a single direction along which it diffuses, as in the heat equation. An elliptic equation has none at all in the real plane — its characteristics are complex — which is why Laplace's equation has no preferred direction of travel and no signals that propagate.

Bending the axes: the three canonical forms

Now we cash in. We introduce new variables — characteristic coordinates — built from the characteristic curves themselves, and rewrite the equation in them. The payoff is that the cross term and one or two of the pure second derivatives vanish, leaving the canonical form for that type. Each type has its own template, and recognising it is the goal of the whole exercise.

Hyperbolic:   u_(xi eta) = (lower-order terms)        [two real families xi, eta]
              equivalently  u_(tau tau) - u_(sigma sigma) = ...   (wave-equation shape)

Parabolic:    u_(eta eta) = (lower-order terms)        [one family; xi rides along)
              heat-equation shape:  u_t = k u_xx

Elliptic:     u_(alpha alpha) + u_(beta beta) = (lower-order terms)
              Laplace shape:  u_xx + u_yy = 0

Read these as the same three faces you already know. The hyperbolic canonical form u_(xi eta) = ... is, after a 45-degree turn, exactly u_tt - c^2 u_xx = ... — the wave equation skeleton, whose solutions split into left- and right-movers u(x,t) = F(x - c t) + G(x + c t) sliding along the two characteristic families. The parabolic form is the heat-equation skeleton u_t = k u_xx. The elliptic form is the Laplace skeleton u_xx + u_yy = 0. Classification is, quite literally, the statement that every second-order linear equation is one of these three in disguise.

A worked reduction, step by step

Let us turn the recipe on a concrete hyperbolic equation and watch the canonical form fall out by hand. Take u_xx - u_yy = 0 — already simple, but the mechanics are identical for messier ones, and seeing them on a transparent case is the best way to trust the method.

1. Read off A = 1, 2B = 0, C = -1. The discriminant is B^2 - A C = 0 - (1)(-1) = 1 > 0, so the equation is hyperbolic and we expect two real characteristic families.
2. Solve the characteristic equation A (dy/dx)^2 - 2B (dy/dx) + C = 0, i.e. (dy/dx)^2 - 1 = 0. The two slopes are dy/dx = +1 and dy/dx = -1.
3. Integrate each slope into a family of lines: y - x = constant and y + x = constant. These two families are the characteristics; their constant-labels become the new coordinates xi = x + y and eta = x - y.
4. Change variables with the chain rule. The pure second derivatives recombine and the equation u_xx - u_yy = 0 becomes u_(xi eta) = 0 — the cross term is now the only survivor.
5. Solve the canonical form directly. u_(xi eta) = 0 says u is a sum of a function of xi alone and a function of eta alone: u = F(xi) + G(eta) = F(x + y) + G(x - y). Two arbitrary functions, one per characteristic family — the d'Alembert pattern in plain sight.

Notice what the canonical form gave us for free: the general solution of a second-order PDE carries two arbitrary functions, not arbitrary constants. This is the running theme of the whole subject — a PDE's general solution is parametrised by functions, and it is the data you impose (here, values and a normal derivative along a non-characteristic curve) that pins those functions down. The number of characteristic families and the canonical form together tell you exactly how much freedom the data must remove.

Why the canonical form decides who governs

The canonical form is not cosmetic bookkeeping — it is the reason the type controls everything that comes next. Once an equation is laid bare as the wave, heat, or Laplace skeleton, it inherits that skeleton's entire personality: which data is well-posed, how fast information moves, whether solutions smooth or stay rough. The next guide pursues that story in full; here is the bridge.

The characteristics drawn in this guide are literally the answer to "how fast does information travel?". Hyperbolic equations have real characteristics, so signals ride along them at finite speed — this is finite propagation speed, the reason a plucked string's disturbance reaches your ear after a delay, not instantly. Parabolic equations smear along their single family with infinite speed — flip a heat source on and, mathematically, every point feels it at once, though faintly. Elliptic equations have no real characteristics at all, so there is no propagation to speak of: an elliptic solution is an instantaneous equilibrium where every point depends on every other.

And the same characteristics decide what data you are allowed to give. The honest warning to carry forward: matching the canonical form to a problem is not optional flourish. Hand Cauchy data to an elliptic equation like Laplace's and the problem is ill-posed — tiny wiggles in the data blow up without bound. Run the heat equation backward in time and it is ill-posed too. Each type comes with its own list of well-posed problems, and the canonical form is exactly the lens that reveals which list you are on.