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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.
Fabert, Oliver 1 ; Lamoree, Niek 1
@article{10_2140_agt_2023_23_461,
author = {Fabert, Oliver and Lamoree, Niek},
title = {Time-periodic solutions of {Hamiltonian} {PDEs} using pseudoholomorphic curves},
journal = {Algebraic and Geometric Topology},
pages = {461--508},
publisher = {mathdoc},
volume = {23},
number = {1},
year = {2023},
doi = {10.2140/agt.2023.23.461},
url = {http://geodesic.mathdoc.fr/articles/10.2140/agt.2023.23.461/}
}
TY - JOUR AU - Fabert, Oliver AU - Lamoree, Niek TI - Time-periodic solutions of Hamiltonian PDEs using pseudoholomorphic curves JO - Algebraic and Geometric Topology PY - 2023 SP - 461 EP - 508 VL - 23 IS - 1 PB - mathdoc UR - http://geodesic.mathdoc.fr/articles/10.2140/agt.2023.23.461/ DO - 10.2140/agt.2023.23.461 ID - 10_2140_agt_2023_23_461 ER -
%0 Journal Article %A Fabert, Oliver %A Lamoree, Niek %T Time-periodic solutions of Hamiltonian PDEs using pseudoholomorphic curves %J Algebraic and Geometric Topology %D 2023 %P 461-508 %V 23 %N 1 %I mathdoc %U http://geodesic.mathdoc.fr/articles/10.2140/agt.2023.23.461/ %R 10.2140/agt.2023.23.461 %F 10_2140_agt_2023_23_461
Fabert, Oliver; Lamoree, Niek. Time-periodic solutions of Hamiltonian PDEs using pseudoholomorphic curves. Algebraic and Geometric Topology, Tome 23 (2023) no. 1, pp. 461-508. doi: 10.2140/agt.2023.23.461
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