The paper “Validity of the hyperbolic Whitham modulation equations in Sobolev spaces“, co-authored by Tom Bridges, Anna Kostianko, and Sergey Zelik, has been published in the Journal of Differential Equations. The paper proves that modulation of periodic wave trains, of the defocusing nonlinear Schrödinger equation, can be approximated by solutions of the Whitham modulation equations, in the hyperbolic case, on a natural time scale. A link to the published paper is here. A link to the final form arXiv version is here. The screenshot below shows one of the main theorems.