Orateur
Description
Typical solutions of small dispersion NLS regularize the shock formation predicted by the dispersionless equation in a universal way. This behavior, described in detail by Bertola and Tovbis, describes solutions which remain bounded in a neighborhood of the dispersionless shock point. In this talk I will discuss a special class of initial data, introduced by Talanov, for which the dispersionless shock generates finite time blow-up. Our results show this blow-up is regularized by dispersion, but have amplitudes inversely proportional to the dispersion. This new universal behavior is described by a particular solution of the Painleve III equation associated with "rogue waves of infinite order". We further show this blow-up behavior extends to the full focusing NLS hierarchy. This is joint work with Peter Miller and Robbie Buckingham.