**Author:**
Guido Gentile and Michela Procesi

**Title:** *
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.

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

SISSA

Trieste, I-34014, Italy

e-mail: procesi@ma.sissa.it