Birkhoff Normal Form with Application to Gross Pitaevskii Equation

Abstract

<p>This thesis investigates a 1-dimensional Gross-Pitaevskii (GP) equation from the viewpoint of a system of Hamiltonian partial differential equations (PDEs). A theorem on Birkhoff normal forms is a particularly important goal of this study. The resulting system is a perturbed system of a completely resonant system, which we analyze, using several forms of perturbation theory.</p> <p>In chapter two, we study estimates 011 integrals of products of four Hermite functions, which represent coefficients of mode coupling, and play an important role in the proof of the Birkhoff normal form theorem. This is a basic problem, which has a close relationship with a problem of Besicovitch, namely the behavior of the L^p norms of L² -normalized Hermite functions.</p> <p>In chapter three we carefully reconsider the linear Schrodinger equation with a harmonic potential, and we introduce a family of Hilbert spaces for studying the GP equation, which generalize the traditional energy spaces in which one works. One unexpected fact is that these function spaces have a close relationship with the former works for the tempered distributions, in particular the N-representation theory due to B. Simon, and V. Bargmann's theory, which uncovers relationship between the tempered distributions and his function spaces through the so-called Segal-Bargmann transformation. In addition, our function spaces have a nice relationship with the Sobolev spaces. In this chapter, a few other questions regarding these function spaces are discussed.</p> <p>In chapter four the proof of the Birkhoff normal form theorem on spaces we have introduced are provided. The analysis is divided into two cases according to the regularity of the related function space. After proving the Birkhoff normal form theorem, we made an analysis of the impact of the perturbation on the main part of the GP system, which we remark is completel:y resonant.</p>