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• W1647649584 abstract "Consider the focussing cubic nonlinear Schrodinger equation in $R^3$: $$ ipsi_t+Deltapsi = -|psi|^2 psi. $$ It admits special solutions of the form $e^{italpha}phi$, where $phi$ is a Schwartz function and a positive ($phi>0$) solution of $$ -Delta phi + alphaphi = phi^3. $$ The space of all such solutions, together with those obtained from them by rescaling and applying phase and Galilean coordinate changes, called standing waves, is the eight-dimensional manifold that consists of functions of the form $e^{i(v cdot + Gamma)} phi(cdot - y, alpha)$. We prove that any solution starting sufficiently close to a standing wave in the $Sigma = W^{1, 2}(R^3) cap |x|^{-1}L^2(R^3)$ norm and situated on a certain codimension-one local Lipschitz manifold exists globally in time and converges to a point on the manifold of standing waves. Furthermore, we show that $mc N$ is invariant under the Hamiltonian flow, locally in time, and is a centre-stable manifold in the sense of Bates, Jones. The proof is based on the modulation method introduced by Soffer and Weinstein for the $L^2$-subcritical case and adapted by Schlag to the $L^2$-supercritical case. An important part of the proof is the Keel-Tao endpoint Strichartz estimate in $R^3$ for the nonselfadjoint Schrodinger operator obtained by linearizing around a standing wave solution." @default.
• W1647649584 created "2016-06-24" @default.
• W1647649584 creator A5058437827 @default.
• W1647649584 date "2008-03-15" @default.
• W1647649584 modified "2023-10-14" @default.
• W1647649584 title "A Centre-Stable Manifold for the Focussing Cubic NLS in $${mathbb{R}}^{1+3}$$" @default.
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• W1647649584 doi "https://doi.org/10.1007/s00220-008-0427-3" @default.
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