@article{22045,
  abstract     = {We consider the initial-value problem for the cubic-quintic nonlinear Schrödinger equation (𝑖𝜕𝑡+Δ)⁢𝜓 =𝛼1⁢𝜓 −𝛼3⁢|𝜓|2⁢𝜓 +𝛼5⁢|𝜓|4⁢𝜓 in three spatial dimensions in the class of solutions with |𝜓⁡(𝑥)| →𝑐 >0 as |𝑥| →∞. Here 𝛼1, 𝛼3, 𝛼5, and 𝑐 are such that 𝜓⁡(𝑥) ≡𝑐 is an energetically stable equilibrium solution to this equation. Normalizing the boundary condition to 𝜓⁡(𝑥) →1 as |𝑥| →∞, we study the associated initial-value problem for 𝑢 =𝜓 −1 and prove a scattering result for small initial data in a weighted Sobolev space.},
  author       = {Killip, Rowan and Murphy, Jason and Visan, Monica},
  issn         = {0036-1410},
  journal      = {SIAM Journal on Mathematical Analysis},
  keywords     = {cubic-quintic NLS, nonvanishing boundary conditions, space-time resonances, scattering},
  number       = {3},
  pages        = {2681--2739},
  publisher    = {Society for Industrial & Applied Mathematics},
  title        = {{The initial-value problem for the cubic-quintic NLS with nonvanishing boundary conditions}},
  doi          = {10.1137/17m1116702},
  volume       = {50},
  year         = {2018},
}