Birkhoff normal form for the nonlinear Schrödinger equation

Abstract

This paper is intended to highlight the differences between the nonlinear Schrödinger equation (NLS) posed on a compact manifold (such as a torus ${\mathbb{T}}^d$) in contrast to being posed on noncompact regions such as on all of ${\mathbb{R}}^d$. The point is to indicate a number of specific facts about the behavior of solutions in the former situation, in which they have the possibility for recurrence, and the latter, in which solutions have the tendency to disperse. This is the topic of the short article by McKean [8], in which the issue of resonance for partial differential evolution equations is discussed. The aspect of this question that we describe in the present paper is that there are different normal forms for these two cases, which rephrases the question as to which of the nonlinear terms are the resonant terms, and what is the appropriate Birkhoff normal form for the NLS. We show that, at least in a neighborhood of zero of an appropriate Hilbert space, the fourth order Birkhoff normal form transformation for the NLS equation is able to eliminate all of the nonresonant terms of the Hamiltonian, and as well, all of the resonant terms. The result is a prognosis, to the negative, for the formal theory of wave turbulence for Hamiltonian partial differential equations posed in Sobolev spaces over ${\mathbb{R}}^d$.