Instability of an integrable nonlocal NLS

Abstract

In this note, we discuss the global dynamics of an integrable nonlocal NLS on R, which has been the object of a recent investigation by integrable systems methods. We prove two results that are in striking contrast with the case of the local cubic focusing NLS. First, finite-time blow-up solutions exist with arbitrarily small initial data in Hs(R), for any s⩾0. On the other hand, the solitons of the local NLS, which are also solutions to the nonlocal equation, are unstable by blow-up for the latter.