Sonin's argument, the shape of solitons, and the most stably singular matrix

Abstract

We present two adaptations of an argument of Sonin, which is known to be a powerful tool for obtaining both qualitative and quantitative information about special functions. Our particular applications are as follows: (i) We give a rigorous formulation and proof of the following assertion about focusing NLS in any dimension: The spatial envelope of a spherically symmetric soliton in a repulsive potential is a non-increasing function of the radius. (ii) Driven by the question of determining the most stably singular matrix, we determine the location of the maximal eigenvalue density of an $n\times n$ GUE matrix. Strikingly, in even dimensions, this maximum is \emph{not} at zero.