The semiclassical limit of the defocusing NLS hierarchy

We establish the semiclassical limit of the one-dimensional defocusing cubic nonlinear Schrödinger (NLS) equation. Complete integrability is exploited to obtain a global characterization of the weak limits of the entire NLS hierarchy of conserved densities as the field evolves from reflectionless initial data under all the associated commuting flows. Consequently, this also establishes the zero-dispersion limit of the modified Korteweg-de Vries equation that resides in that hierarchy. We have adapted and clarified the strategy introduced by Lax and Levermore to study the zero-dispersion limit of the Korteweg-de Vries equation, expanding it to treat entire integrable hierarchies and strengthening the limits obtained. A crucial role is played by the convexity of the underlying log-determinant with respect to the times associated with the commuting flows.