the Cauchy data
/ Cauchy -> koh-SHEE /
When you launch a ball, you specify two things at the start: where it is and how fast it is going. The Cauchy data is the PDE version of exactly that — the full bundle of starting information you lay down on a surface so that a higher-order equation has a determined evolution. It is named after Augustin-Louis Cauchy, who framed the general 'give the function and its derivatives on a slice, then march outward' problem. For an evolution equation the slice is the line t = 0; more generally it can be any curve or surface in the domain.
Precisely: for an equation of order m, the Cauchy data on a surface S is the value of the unknown u together with its normal derivatives up to order m - 1 on S. For a second-order equation that means u and its first normal derivative — for the wave equation u_tt = c^2 u_xx, that is u(x, 0) = f(x) and u_t(x, 0) = g(x), the displacement and velocity. The phrase 'Cauchy problem' means: solve the PDE given Cauchy data on an initial surface, with no other walls or boundaries in sight — an evolution that runs outward from the data slice rather than being penned inside a box.
There is a crucial catch that separates Cauchy data from mere boundary data: the surface must be non-characteristic. If the surface you place your data on happens to be a characteristic of the equation, the data and the equation conflict or fail to determine the next layer, and you cannot march forward. And the type of the equation decides whether a Cauchy problem is even sensible: posing Cauchy data is the right thing for hyperbolic and parabolic equations, but for an elliptic equation like Laplace's it is famously ill-posed — Hadamard's warning that the same words can be well-posed or catastrophic depending on the equation.
For the wave equation u_tt = c^2 u_xx, the Cauchy data on the line t = 0 is the pair u(x, 0) = f(x) (where the string is) and u_t(x, 0) = g(x) (how fast each point moves). d'Alembert's formula then writes the unique solution directly from this pair as a sum of a right-mover and a left-mover plus an integral of g.
Cauchy data for a second-order equation = the function plus its first normal derivative on the surface.
Cauchy data is not automatically well-posed: prescribing it on a characteristic surface, or for an elliptic equation, can leave the problem unsolvable or wildly unstable — the surface must be non-characteristic and the equation of the right type.