  The fact that the Korteweg-de-Vries equation offers a good approximation of
long-wave solutions of small amplitude to the one-dimensional Gross-Pitaevskii
equation was derived several years ago in the physical literature. In this
paper, we provide a rigorous proof of this fact, and compute a precise estimate
for the error term. Our proof relies on the integrability of both the
equations. In particular, we give a relation between the invariants of the two
equations, which, we hope, is of independent interest.
