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.


Mathematics and Statistics


The work of the first author was partially supported by a grant from the Simons Foundation (342360) and by NSF grants DMS-1265868 and DMS-1600942. The work of the second author was supported by NSF Postdoctoral Fellowship DMS-1400706 at the University of California, Berkeley. The work of the third author was supported by NSF grant DMS-1500707. Part of the work on this project was supported by the NSF grant DMS-1440140 while the authors were in residence at the Mathematical Sciences Research Institute in Berkeley, California, during the fall 2015 semester.

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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Article - Journal

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Publication Date

01 May 2018