@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},
}

@article{22051,
  abstract     = {We construct solutions with prescribed scattering state to the cubic-quintic NLS (mathematical formular)in three spatial dimensions in the class of solutions with (mathematical formular). This models disturbances in an infinite expanse of (quantum) fluid in its quiescent state— the limiting modulus c corresponds to a local minimum in the energy density.
Our arguments build on work of Gustafson, Nakanishi, and Tsai on the (defocusing) Gross–Pitaevskii equation. The presence of an energy-critical nonlinearity and changes in the geometry of the energy
functional add several new complexities. One new ingredient in our argument is a demonstration that
solutions of such (perturbed) energy-critical equations exhibit continuous dependence on the initial data
with respect to the weak topology on H1/x.},
  author       = {Killip, Rowan and Murphy, Jason and Visan, Monica},
  issn         = {1948-206X},
  journal      = {Analysis & PDE},
  keywords     = {final-state problem, wave operators, cubic-quintic NLS, nonvanishing boundary conditions},
  number       = {7},
  pages        = {1523--1574},
  publisher    = {Mathematical Sciences Publishers},
  title        = {{The final-state problem for the cubic-quintic NLS with nonvanishing boundary conditions}},
  doi          = {10.2140/apde.2016.9.1523},
  volume       = {9},
  year         = {2016},
}