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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

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.

Hamiltonian PDE, pseudoholomorphic curve, small divisor problem, Gromov compactness, Diophantine approximation
Mathematical Subject Classification
Primary: 35B10, 37K58, 53D40, 58E30
Received: 8 July 2021
Accepted: 11 October 2021
Published: 27 March 2023
Oliver Fabert
Department of Mathematics
Vrije Universiteit Amsterdam
Niek Lamoree
Department of Mathematics
Vrije Universiteit Amsterdam

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