The Final-State Problem for the Cubic-Quintic NLS with Nonvanishing Boundary Conditions
We construct solutions with prescribed scattering state to the cubic-quintic NLS (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| → ∞. 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 Hx1.
R. Killip et al., "The Final-State Problem for the Cubic-Quintic NLS with Nonvanishing Boundary Conditions," Analysis and PDE, vol. 9, no. 7, pp. 1523 - 1574, Mathematical Sciences Publishers (MSP), Nov 2016.
The definitive version is available at https://doi.org/10.2140/apde.2016.9.1523
Mathematics and Statistics
Keywords and Phrases
Cubic-quintic NLS; Final-state problem; Nonvanishing boundary conditions; Wave operators
International Standard Serial Number (ISSN)
Article - Journal
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01 Nov 2016