FM 2007-03 (mp_arc 07-94; arXiv:0704.2534v1 [math.AP])

Author: Guido Gentile and Michela Procesi

Title: Periodic solutions for the Schrödinger equation with nonlocal smoothing nonlinearities in higher dimension
 
Abstract: We consider the nonlinear Schrödinger equation in higher dimension with Dirichlet boundary conditions and with a non-local smoothing nonlinearity. We prove the existence of small amplitude periodic solutions. In the fully resonant case we find solutions which at leading order are wave packets, in the sense that they continue linear solutions with an arbitrarily large number of resonant modes. The main difficulty in the proof consists in solving a "small divisor problem" which we do by using a renormalisation group approach.

Keywords: Nonlinear Schrödinger equation; Higher dimension; Periodic solutions; Renormalization Group; Lindstedt series method; Diophantine conditions; Tree formalism; Lyapunov-Schmidt decomposition.

Guido Gentile
Dipartimento di Matematica
Universitā di Roma Tre
Largo San Leonardo Murialdo 1, 00146 Roma, Italy
e-mail: gentile@mat.uniroma3.it

Michela Procesi
Dipartimento di Matematica
Universitā di Roma Tre
Largo San Leonardo Murialdo 1, 00146 Roma, Italy
e-mail: procesi@mat.uniroma3.it