Related papers: Birkhoff Normal Form and Hamiltonian PDEs

We consider {\em discretized} Hamiltonian PDEs associated with a Hamiltonian function that can be split into a linear unbounded operator and a regular nonlinear part. We consider splitting methods associated with this decomposition. Using a…

We prove an abstract Birkhoff normal form theorem for Hamiltonian partial differential equations on torus. The normal form is complete up to arbitrary finite order. The proof is based on a valid non-resonant condition and a suitable norm of…

Birkhoff normal form is a power series expansion associated with the local behavior of the Hamiltonian systems near a critical point. It is known to be convergent for integrable systems under some non-degeneracy conditions. By means of an…

The multi-symplectic form for Hamiltonian PDEs leads to a general framework for geometric numerical schemes that preserve a discrete version of the conservation of symplecticity. The cases for systems or PDEs with dissipation terms has…

We prove an abstract Birkhoff normal form theorem for Hamiltonian Partial Differential Equations. The theorem applies to semilinear equations with nonlinearity satisfying a property that we call of Tame Modulus. Such a property is related…

We consider Hamiltonian PDEs that can be split into a linear unbounded operator and a regular non linear part. We consider abstract splitting methods associated with this decomposition where no discretization in space is made. We prove a…

We consider an undamped nonlinear hinged-hinged beam with stretching nonlinearity as an infinite dimensional hamiltonian system. We obtain analytically a quantitative Birkhoff Normal Form, via a nonlinear coordinate transformation that…

We study several aspects of the regular deformations of completely integrable systems. Namely, we prove the existence of a Hamiltonian normal form for these deformations and we show the necessary and sufficient conditions a perturbation has…

Birkhoff normal forms are commonly used in order to ensure the so called "effective stability" in the neighborhood of elliptic equilibrium points for Hamiltonian systems. From a theoretical point of view, this means that the eventual…

This paper combines the decay of high modes with the smallness introduced by high orders, leading to a normal form lemma for infinite-dimensional Hamiltonian systems under ultra-differentiable regularity. We prove the sub-exponential…

In this paper we give an extension of the Birkhoff--Lewis theorem to some semilinear PDEs. Accordingly we prove existence of infinitely many periodic orbits with large period accumulating at the origin. Such periodic orbits bifurcate from…

We derive an explicit tree based ansatz for the Birkhoff normal form up to any order in the context of Hamiltonian PDEs. To do so we make use of a tree based representation of iterated Poisson brackets to encode the nested Taylor expansions…

This article is concerned with analytic Hamiltonian dynamical systems in infinite dimension in a neighborhood of an elliptic fixed point. Given a quadratic Hamiltonian, we consider the set of its analytic higher order perturbations. We…

We study the general structure of formal perturbative solutions to the Hamiltonian perturbations of spatially one-dimensional systems of hyperbolic PDEs. Under certain genericity assumptions it is proved that any bihamiltonian perturbation…

Our main goal is the comparative study of singularities of solutions to the systems of first order quasilinear PDEs and their perturbations containing higher derivatives. The study is focused on the subclass of Hamiltonian PDEs with one…

Consider a linear autonomous Hamiltonian system with a time periodic bound state solution. In this paper we study the structural instability of this bound state ^M relative to time almost periodic perturbations which are small, localized…

In this paper we consider an abstract class of quasi-linear para-differential equations on the circle. For each equation in the class we prove the existence of a change of coordinates which conjugates the equation to a diagonal and constant…

The aim of this paper is to introduce a class of Hamiltonian autonomous systems in dimension 4 which are completely integrable and their dynamics is described in all details. They have an equilibrium point which is stable for some rare…

Consider a general linear Hamiltonian system $\partial_{t}u=JLu$ in a Hilbert space $X$. We assume that$\ L: X \to X^{*}$ induces a bounded and symmetric bi-linear form $\left\langle L\cdot,\cdot\right\rangle $ on $X$, which has only…

The past few years have witnessed an increased interest in learning Hamiltonian dynamics in deep learning frameworks. As an inductive bias based on physical laws, Hamiltonian dynamics endow neural networks with accurate long-term…