Author: Guido Gentile and Michela Procesi
Conservation of resonant periodic solutions for the
one-dimensional nonlinear Schrödinger equation
Periodic solutions for completely resonant nonlinear wave equations
Abstract: We consider the one-dimensional nonlinear Schrödinger equation with Dirichlet boundary conditions in the fully resonant case (absence of the zero-mass term). We investigate conservation of small amplitude periodic-solutions for a large measure set of frequencies. In particular we show that there are infinitely many periodic solutions which continue the linear ones involving an arbitrary number of resonant modes, provided the corresponding frequencies are large enough, say greater than a certain threshold value depending on the number of resonant modes. If the frequencies of the latter are close enough to such a threshold, then they can not be too distant from each other. Hence we can interpret such solutions as perturbations of wave packets with large wave number.
Keywords: Nonlinear Schrödinger equation; Periodic solutions; Resonances; Lindstedt series method; Tree formalism; Lyapunov-Schmidt decomposition; Renormalization Group; Diophantine conditions.
Dipartimento di Matematica
Universitā di Roma Tre
Largo San Leonardo Murialdo 1, 00146 Roma, Italy
Trieste, I-34014, Italy