Sonin's argument, the shape of solitons, and the most stably singular matrix (Harmonic Analysis and Nonlinear Partial Differential Equations)

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; see [12]. 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 aspherically symmetric soliton in arepulsive 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 ncross n GUE matrix. Strikingly, in even dimensions, this maximum is not at zero.