Solitons and scattering for the cubic-quintic nonlinear Schrödinger equation on ℝ^3

Rowan Killip, Tadahiro Oh, Oana Pocovnicu, Monica Visan

Research output: Contribution to journalArticlepeer-review


We consider the cubic-quintic nonlinear Schrödinger equation:

i∂_t u = −∆u − |u|^2u + |u|^4u.

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 this region is partially bounded by solitons but 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.
Original languageEnglish
Pages (from-to)469-548
Number of pages80
JournalArchive for Rational Mechanics and Analysis
Issue number1
Early online date24 Mar 2017
Publication statusPublished - Jul 2017


  • Energy-critical NLS
  • cubic-quintic NLS
  • scattering
  • soliton
  • Gagliardo–Nirenberg inequality


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