Scattering for the non-radial 3D cubic nonlinear Schroedinger equation

By Thomas Duyckaerts, Justin Holmer, Svetlana Roudenko. 2007

274 citations recorded by the source corpus.

Scattering of radial $H^1$ solutions to the 3D focusing cubic nonlinear Schr\"odinger equation below a mass-energy threshold $M[u]E[u] < M[Q]E[Q]$ and satisfying an initial mass-gradient bound $\|u_0\|_{L^2} \|\nabla u_0 \|_{L^2} < \|Q\|_{L^2} \|\nabla Q\|_{L^2}$, where $Q$ is the ground state, was established in Holmer-Roudenko (2007). In this note, we extend the result in Holmer-Roudenko (2007) to non-radial $H^1$ data. For this, we prove a non-radial profile decomposition involving a spatial translation parameter. Then, in the spirit of Kenig-Merle (2006), we control via momentum conservation the rate of divergence of the spatial translation parameter and by a convexity argument based on a local virial identity deduce scattering. An application to the defocusing case is also mentioned.

Subfields: Mathematical Physics, Nuclear and High Energy Physics, Statistical and Nonlinear Physics

View Paper: https://www.semanticscholar.org/paper/5ab9ec083a6ee1b5cd65efde369eed37428ae06f

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Citations in the corpus

Scattering threshold for the focusing nonlinear Klein–Gordon equation (2010, background): “the critical case), except for the NLS with a subcritical power [15, 4].”
On well posedness for the inhomogeneous nonlinear Schr\"odinger equation (2016, background): “1), does not give sufficient tools to study other interesting questions, for instance, scattering and blow up investigated by Kenig-Merle [21], Holmer-Roudenko-Duyckaerts [9] and others, for the NLS equation.”
A new proof of scattering below the ground state for the 3d radial focusing cubic NLS (2016, background): “The results of [3, 5] were generalized to other dimensions and intercritical nonlinearities by [2, 4].”
The focusing energy-critical nonlinear Schr\"odinger equation in dimensions five and higher (2008, methodology): “This mimics similar arguments in [10], [19] and relies crucially on the L2 x-compactness properties proved in the first step.”
Nondispersive radial solutions to energy supercritical non-linear wave equations, with applications (2008, methodology): “For instance, it was applied to the “mass-critical” NLS in the radial case, in dimension d ≥ 2, in both focusing and defocusing cases, in works of Killip, Tao, Vişan and Zhang, [38], [21], [23], to the focusing cubic NLS for d = 3, by Holmer–Roudenko [13] and Duyckaerts–Holmer–Roudenko [6], to corot”