On the rigidity of solitary waves for the focusing mass-critical NLS in dimensions d≥2

Abstract

For the focusing mass-critical NLS iu t + Δu = -{pipe}u{pipe} 4/du, it is conjectured that the only global non-scattering solution with ground state mass must be a solitary wave up to symmetries of the equation. In this paper, we settle the conjecture for H x1 initial data in dimensions d = 2, 3 with spherical symmetry and d ≥ 4 with certain splitting-spherically symmetric initial data. © 2012 Science China Press and Springer-Verlag Berlin Heidelberg.