On the Korteweg–de Vries Long-Wave Approximation of the Gross–Pitaevskii Equation I

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 (see e.g. [17]). 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.