Global Dynamics below the standing waves for the focusing semilinear   Schr\"{o}dinger equation with a repulsive Dirac delta potential
  We consider the focusing mass supercritical semilinear Schr\"{o}dinger equation with a repulsive Dirac delta potential on the real line (deltaNLS). Our aim in the present paper is to find a necessary and sufficient condition on the data below the standing wave of NLS without a potential to determine the global behavior of the solution. Moreover, we determine the global dynamics of the radial solution whose mass-energy is larger than that of the standing wave and smaller than that of the standing wave of deltaNLS. 
Masahiro Ikeda, Takahisa Inui
math.AP