Time-periodic solutions of Hamiltonian PDEs using pseudoholomorphic curves

Fabert, Oliver; Lamoree, Niek

doi:10.2140/agt.2023.23.461

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Time-periodic solutions of Hamiltonian PDEs using pseudoholomorphic curves

Oliver Fabert and Niek Lamoree

Algebraic & Geometric Topology 23 (2023) 461–508

DOI: 10.2140/agt.2023.23.461

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.

Keywords

Hamiltonian PDE, pseudoholomorphic curve, small divisor problem, Gromov compactness, Diophantine approximation

Mathematical Subject Classification

Primary: 35B10, 37K58, 53D40, 58E30

References

Publication

Received: 8 July 2021

Accepted: 11 October 2021

Published: 27 March 2023

Authors

Oliver Fabert
Department of Mathematics Vrije Universiteit Amsterdam Amsterdam Netherlands

Niek Lamoree
Department of Mathematics Vrije Universiteit Amsterdam Amsterdam Netherlands

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Volume 23, issue 1 (2023)