# Time-periodic solutions of Hamiltonian PDEs using pseudoholomorphic curves

## TL;DR

This paper extends pseudoholomorphic curve methods from Floer theory to infinite-dimensional Hamiltonian PDEs, proving existence and multiplicity of time-periodic solutions under regularizing nonlinearities.

## Contribution

It introduces an infinite-dimensional Gromov-Floer compactness framework and establishes a cup-length estimate for periodic solutions in certain Hamiltonian PDEs.

## Key findings

- Existence of forced time-periodic solutions for Hamiltonian PDEs with regularizing nonlinearities.
- Development of an infinite-dimensional Gromov-Floer compactness theorem.
- A lower bound on the number of periodic solutions based on cup-length.

## Abstract

We extend the pseudoholomorphic curve methods from Floer theory to infinite-dimensional phase spaces and use our results to prove the existence of a forced time-periodic solution to a general Hamiltonian PDE with regularizing nonlinearity. In particular, when the nonlinearity is sufficiently regularizing, bounded and time-periodic, we prove an infinite-dimensional version of Gromov-Floer compactness by using ideas from the theory of Diophantine approximations to overcome the small divisor problem. Furthermore, in the case when the infinite-dimensional phase space is a product of a finite-dimensional closed symplectic manifold with linear symplectic Hilbert space, we prove a cup-length estimate for the number of periodic solutions.