The semiclassical limit of the defocusing NLS hierarchy

We establish the semiclassical limit of the one-dimensional defocusing cubi c nonlinear Schrodinger (NLS) equation. Complete integrability is exploited to obtain a global characterization of the weak limits of the entire NLS h ierarchy of conserved densities as the field evolves from reflectionless in itial data under all the associated commuting flows. Consequently, this als o establishes the zero-dispersion limit of the modified Korteweg-de Vries e quation that resides in that hierarchy. We have adapted and clarified the s trategy 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 tim es associated with the commuting flows. (C) 1999 John Wiley & Sons, Inc.