Resonant decompositions and the Imethod for cubic nonlinear Schrodinger on R^2
Abstract
The initial value problem for the cubic defocusing nonlinear Schrödinger equation $i \partial_t u + \Delta u = u^2 u$ on the plane is shown to be globally wellposed for initial data in $H^s (\R^2)$ provided $s>1/2$. The proof relies upon an almost conserved quantity constructed using multilinear correction terms. The main new difficulty is to control the contribution of resonant interactions to these correction terms. The resonant interactions are significant due to the multidimensional setting of the problem and some orthogonality issues which arise.
 Publication:

arXiv eprints
 Pub Date:
 April 2007
 arXiv:
 arXiv:0704.2730
 Bibcode:
 2007arXiv0704.2730C
 Keywords:

 Mathematics  Analysis of PDEs;
 Mathematical Physics;
 35Q55