Inverse scattering on the line for a generalized nonlinear Schroedinger   equation
  A one-dimensional generalized nonlinear Schroedinger equation is considered, and the corresponding inverse scattering problem is analyzed when the potential is compactly supported and depends on the wave function. The unique recovery of the potential is established from an appropriate set of scattering data. 
Tuncay Aktosun, Vassilis G. Papanicolaou, Vassilis Zisis