Validity of the hyperbolic Whitham modulation equations in Sobolev spaces

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