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The final-state problem
for the cubic-quintic NLS with nonvanishing boundary
conditions
Rowan Killip, Jason Murphy and Monica Visan |
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Vol. 9 (2016), No. 7, 1523–1574
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DOI: 10.2140/apde.2016.9.1523
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Abstract |
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We construct solutions with prescribed scattering state to the cubic-quintic NLS in three spatial dimensions in the class of solutions with as . This models disturbances in an infinite expanse of (quantum) fluid in its quiescent state — the limiting modulus 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 . |
Keywords
final-state problem, wave operators, cubic-quintic NLS,
nonvanishing boundary conditions
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Mathematical Subject Classification 2010
Primary: 35Q55
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Milestones
Received: 28 July 2015
Revised: 4 May 2016
Accepted: 9 July 2016
Published: 7 November 2016
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Authors |
| Rowan Killip | |
| Department of Mathematics UCLA Los Angeles, CA 90095-1555 United States |
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| Jason Murphy | |
| Department of Mathematics University of California Berkeley, CA 94720-3840 United States |
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| Monica Visan | |
| Department of Mathematics UCLA Los Angeles, CA 90095-1555 United States |
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