  It is proved that modulation in time and space of periodic wave trains, of
the defocussing nonlinear Schr\"odinger equation, can be approximated by
solutions of the Whitham modulation equations, in the hyperbolic case, on a
natural time scale. The error estimates are based on existence, uniqueness, and
energy arguments, in Sobolev spaces on the real line. An essential part of the
proof is the inclusion of higher-order corrections to Whitham theory, and
concomitant higher-order energy estimates.
