arXiv subjects

Niek Lamoree

Publications and source records attributed to Niek Lamoree.

3 recordsLinked to original sources

arXiv2021

Floer homology for Hamiltonian PDEs: Fredholm theory

By coupling a Hamiltonian mechanical system with a linear Hamiltonian field theory one obtains an infinite-dimensional Hamiltonian system with regularizing nonlinearity, where the underlying phase space is given by the product of a finite-dimensional symplectic manifold with an infinite-dimensional linear symplectic Hilbert space. After our compactness results we continue our program for defining a Floer homology theory for this class of infinite-dimensional Hamiltonian systems. Based on the presence of small divisors we introduce a new notion of nondegeneracy for time-periodic solutions which allows us to prove that the linearization of the nonlinear Floer operator is Fredholm when viewed as a map between suitable Sobolev space completions.

math.SG↗

arXiv2021

Cuplength estimates for periodic solutions of Hamiltonian particle-field systems

We consider a natural class of time-periodic infinite-dimensional nonlinear Hamiltonian systems modelling the interaction of a classical mechanical system of particles with a scalar wave field. When the field is defined on a space torus $\mathbb{T}^d=\mathbb{R}^d/(2π\mathbb{Z})^d$ and the coordinates of the particles are constrained to a submanifold $Q\subset\mathbb{T}^d$, we prove that the number of $T$-periodic solutions of the coupled Hamiltonian particle-field system is bounded from below by the $\mathbb{Z}_2$-cuplength of the space of contractible loops in $Q$, provided that the square of the ratio $T/2π$ of time period $T$ and space period $X=2π$ is a Diophantine irrational number. The latter condition is necessary since for the infinite-dimensional version of Gromov-Floer compactness as well as for the $C^0$-bounds we need to deal with small divisors.

math.SG↗

arXiv2021

Time-periodic solutions of Hamiltonian PDEs using pseudoholomorphic curves

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.

math.SG↗