Time-periodic solutions of Hamiltonian PDEs using pseudoholomorphic curves.
We prove the existence of a forced time-periodic solution to general nonlinear Hamiltonian PDEs by using pseudoholomorphic curves as in symplectic homology theory. When the nonlinearity is sufficiently regularizing, bounded and time-periodic we prove that for a generic time period a solution exists, thereby complementing the result by Rabinowitz from 1978. In order to extend the pseudoholomorphic curve methods from Floer theory to our infinite-dimensional setting and prove a type of Gromov-Floer compactness, we use ideas from the theory of Diophantine approximations to overcome the small divisor problem.