@article{22071,
  abstract     = {We consider the cubic–quintic nonlinear Schrödinger equation: (mathematical formular)
In the first part of the paper, we analyze the one-parameter family of ground state solitons associated to this equation with particular attention to the shape of the associated mass/energy curve. Additionally, we are able to characterize the kernel of the linearized operator about such solitons and to demonstrate that they occur as optimizers for a one-parameter family of inequalities of Gagliardo–Nirenberg type. Building on this work, in the latter part of the paper we prove that scattering holds for solutions belonging to the region R of the mass/energy plane where the virial is positive. We show that this region is partially bounded by solitons also by rescalings of solitons (which are not soliton solutions in their own right). The discovery of rescaled solitons in this context is new and highlights an unexpected limitation of any virial-based methodology.},
  author       = {Killip, Rowan and Oh, Tadahiro and Pocovnicu, Oana and Visan, Monica},
  issn         = {1432-0673},
  journal      = {Archive for Rational Mechanics and Analysis},
  pages        = {469--548},
  publisher    = {Springer Nature},
  title        = {{Solitons and scattering for the cubic-quintic nonlinear Schrödinger equation on R^3}},
  doi          = {10.1007/s00205-017-1109-0},
  volume       = {225},
  year         = {2017},
}