| |
MSRI Summer Microprogram on Nonlinear Partial Differential Equations |
| July 23, 2007 to August 10, 2007 |
| Organized By: L. C. Evans (UC Berkeley, Chair), C. Gutierrez (Temple), C. Sogge (Johns Hopkins), D. Tataru (UC Berkeley) |
| |
| Return to Workshop Description |
| |
A Centre-Stable Manifold for the Focussing Cubic NLS in R^{3+1}
Monday August 6, 2007
11:00AM - 12:00PM
Speakers:
Marius Beceanu
Abstract:
Consider the focussing cubic nonlinear Schr\"odinger equation in
$R^3$: $$ i\psi_t+\Delta\psi = -|\psi|^2 \psi. $$ It admits special solutions of the form $e^{it\alpha}\phi$, where $\phi$ is a Schwartz function and a positive ($\phi>0$) solution of $$ -\Delta \phi + \alpha\phi = \phi^3. $$
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 $\Sigma = H^1(R^3) \cap |x|^{-1}L^2(R^3)$ 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 the manifold is invariant under the Hamiltonian flow, locally in time, and is a centre-stable manifold in the sense of Bates, Jones
|
