The Initial-Value Problem for the Cubic-Quintic NLS with Nonvanishing Boundary Conditions
We consider the initial-value problem for the cubic-quintic nonlinear Schrödinger equation (i∂t + Δ)ψ = α1ψ − α3|ψ|2ψ + α5|ψ|4ψ 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.
R. Killip et al., "The Initial-Value Problem for the Cubic-Quintic NLS with Nonvanishing Boundary Conditions," SIAM Journal on Mathematical Analysis, vol. 50, no. 3, pp. 2681 - 2739, Society for Industrial and Applied Mathematics (SIAM), May 2018.
The definitive version is available at https://doi.org/10.1137/17M1116702
Mathematics and Statistics
Keywords and Phrases
Boundary conditions; Initial value problems; Nonlinear equations; Scattering; Sobolev spaces; Dinger equation; Quintic; Space time; Spatial dimension; Stable equilibrium; Weighted Sobolev spaces; C (programming language); Cubic-quintic NLS; Nonvanishing boundary conditions; Scattering; Space-time resonances
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01 May 2018