Floer theory for Hamiltonian PDE using model theory

TL;DR

This paper extends Hamiltonian Floer theory from finite to infinite-dimensional settings using non-standard model theory, applied to Hamiltonian PDEs like the nonlinear Schrödinger equation, under certain natural restrictions.

Contribution

It introduces a novel approach employing non-standard model theory to generalize Floer theory to infinite-dimensional Hamiltonian PDEs without requiring prior knowledge of the theory.

Findings

Floer theory successfully extended to infinite dimensions

Method applies to Hamiltonian PDEs like nonlinear Schrödinger equation

Proof relies on finite-dimensional results without additional assumptions

Abstract

Under natural restrictions it is known that a nonlinear Schr\"odinger equation is a Hamiltonian PDE which defines a symplectic flow on a symplectic Hilbert space preserving the Hilbert norm. When the potential is one-periodic in time and after passing to the projectivization, it makes sense to ask whether the natural analogue of the Arnold conjecture holds. By employing methods from non-standard model theory we show how Hamiltonian Floer theory can be generalized from finite to infinite dimensions. While our proof entirely builds on finite-dimensional results, we do not ask for any prior knowledge of non-standard model theory.