Let us consider the equation:
. The three dots represent terms of lower order.
Let be a surface of arbitrary shape and let the values of and be given on the surface. Is this data sufficient for the reconstruction of the unique solution of the equation (18)? Does the Cauchy problem for equation (18) with the initial data on have a unique solution? The answer will be positive if it is possible to reconstruct all second derivatives on the surface .
Let us introduce a system of functions ,with and with the change of variables:
On the surface all values do vary. That means the given data allows one to calculate all mixed derivatives and second derivatives .Is it possible to calculate ?
Let us express the derivatives with respect to the old variables through the derivatives with respect to the new variables, omitting terms that are not interesting for us, such as:
and so on.
The equation (18) can be rewritten in the form:
All terms denoted by dots are known on the surface ,so the only condition that supplies a possibility to determine is .If , then the Cauchy problem is unsolvable. In this case, surface is a special one. And we call it a characteristic (or characteristic surface).