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Scattering for the nonradial 3D cubic nonlinear Schrödinger equation. (English) Zbl 1171.35472

Summary: Scattering of radial \(H^1\) solutions to the 3D focusing cubic nonlinear Schrödinger 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 [J. Holmer and S. Roudenko, AMRX, Appl. Math. Res. Express 2007, Article ID abm004, 29 p. (2007; Zbl 1130.35361 )]. In this note, we extend the result in J. Holmer and S. Roudenko [Commun. Math. Phys. 282, No. 2, 435–467 (2008; Zbl 1155.35094)] 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 C. E. Kenig and F. Merle [Invent. Math. 166, No. 3, 645–675 (2006; Zbl 1115.35125)], 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.

MSC:

35Q55 NLS equations (nonlinear Schrödinger equations)
35Q51 Soliton equations
35B40 Asymptotic behavior of solutions to PDEs
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