What it means to give data to a PDE
By now you have a working machine: the method of characteristics turns a first-order PDE into ODEs that you integrate along curves. But a machine needs a starting point. With ordinary differential equations you handed over a few numbers — initial conditions — and the general solution settled down to one answer. A PDE asks for something richer, and seeing exactly what it asks for is the heart of this guide.
Here is the crucial shift in worldview, the one we flagged at the end of the last guide. An ODE's general solution carries arbitrary constants, so a handful of numbers nails it down. A PDE's general solution carries arbitrary functions — recall that the transport equation u_t + c u_x = 0 has u(x,t) = f(x - c t) for any differentiable f. A whole function's worth of freedom cannot be fixed by a few numbers; you must supply a whole function's worth of data.
That is what Cauchy data is. You choose a curve in the (x,y)-plane — call it the data curve — and along it you prescribe the value of the unknown u. In the solution-surface picture from the last guide this is one drawn thread floating in space: you have painted the height of the surface all along one curve, and you are asking the PDE to weave the rest of the sheet through that thread.
Why one curve is exactly enough — and why a wrong curve fails
Recall how the method actually builds the surface. Through each point of the data curve you launch a characteristic curve, and that family of characteristics sweeps out the whole solution like the teeth of a comb dragging a sheet into existence. For this to work cleanly, the data curve must cut across the characteristics — it must be a thread laid sideways to the grain, so each comb-tooth touches it once and carries a fresh value outward.
Now picture the disaster. Suppose you accidentally choose the data curve to be a characteristic — laid along the grain instead of across it. Two things go wrong at once, and both are fatal. First, every other characteristic misses your curve entirely, so vast regions of the plane get no data at all; the comb never combs them. The surface is simply not determined there — the problem is underdetermined.
Second, your prescribed values along that curve are not free anyway. A characteristic carries its own internal law — for u_t + c u_x = 0 the value must stay constant along it. If the data you painted disagrees with that law, no solution exists; if it happens to agree, then infinitely many solutions match it. Either no answer or a flood of them: the textbook signature of an ill-posed problem, the opposite of the well-posed problem we want.
The non-characteristic condition, made precise
We can now state the rule that this whole guide is named for. The non-characteristic condition is simply this: the data curve must be transverse to the characteristics — nowhere tangent to one. As long as the curve crosses the grain at every point, the method launches a clean web of characteristics off it and the solution is determined, at least locally near the curve. Lie tangent to a characteristic anywhere, and the determinism breaks at that spot.
There is a clean way to check it with a determinant, and it is worth seeing once. Take a u_x + b u_y = c and a data curve parametrised as x = X(r), y = Y(r). At each point compare the curve's own tangent (X'(r), Y'(r)) with the characteristic direction (a, b). They are parallel — the bad case — exactly when the determinant below vanishes. The condition is therefore that this determinant is non-zero all along the curve.
det | X'(r) a |
| Y'(r) b | = X'(r) b - Y'(r) a =/= 0
(curve tangent) vs (characteristic direction a,b)
zero -> curve is tangent to a characteristic -> ILLEGAL
nonzero -> curve crosses the grain -> good Cauchy dataA tiny worked check. For the transport equation u_t + c u_x = 0 the characteristic direction in the (x,t)-plane is (dx, dt) = (c, 1) — the lines x - c t = const. The honest, well-posed choice of data curve is the line t = 0, the initial profile u(x,0): its tangent points purely in x, the characteristics slope sideways, and they are never parallel, so the determinant never vanishes. By contrast a line of slope x = c t is itself a characteristic — prescribe data there and you have walked straight into the failure of the previous section.
Where the slopes come from — and the honest small print
It is worth understanding why transversality is exactly the condition, not just memorising it. Along the data curve you know u, so you can differentiate it to get the derivative of u in the direction of the curve — the tangential slope comes for free. What you are missing is the slope across the curve, the normal derivative. The PDE a u_x + b u_y = c is one linear equation relating u_x and u_y. Combine it with the tangential slope and you have two equations for the two unknowns u_x, u_y.
And here the non-characteristic condition reappears in its true colours: that two-by-two linear system is solvable for the missing normal slope exactly when its determinant — the very one above — is non-zero. When the curve is non-characteristic, the equation hands you the one derivative the data did not, and the surface can grow outward. When the curve is characteristic, the two equations collapse into one direction; the normal slope is left undetermined, and the surface has nothing to tell it which way to bend.
Stepping back: this is one face of a much bigger law
The pattern you have just seen — that good data must avoid the characteristics — is not a quirk of first-order equations. It is the local engine behind a famous result, the Cauchy–Kovalevskaya theorem, which guarantees a unique analytic solution near a data surface precisely when that surface is non-characteristic (and everything in sight is analytic). The determinant test you computed is the first-order shadow of that theorem's transversality hypothesis.
But hold one honest caveat firmly. Non-characteristic data makes the machinery run; it does not by itself make the problem well-posed in the full sense of depending continuously on the data. The classic warning is Laplace's equation u_xx + u_yy = 0: you can prescribe perfectly smooth Cauchy data on a non-characteristic line and still get a solution that explodes under a vanishingly small wiggle in the data. Existence and uniqueness are not the whole of well-posedness — stability is the third leg, and the type of the equation governs it.
That last thread leads straight out of this rung. For first-order and hyperbolic problems, Cauchy data on a non-characteristic curve is the natural, well-behaved way to pose the problem; for elliptic equations like Laplace's it is the wrong kind of data, and a boundary-value problem is what the geometry wants instead. Carry one sentence forward: data is what selects a solution, but which data is legal is dictated jointly by the characteristics and the type of the equation. The next guide watches what happens when the characteristics, even from legal data, finally cross.