Related papers: Birkhoff normal form and splitting methods for sem…
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 a class of Hamiltonian PDEs that can be split into a linear unbounded operator and a regular non linear part, and we analyze their numerical discretizations by symplectic methods when the initial value is small in Sobolev norms.…
We consider a wide class of semi linear Hamiltonian partial differential equa- tions and their approximation by time splitting methods. We assume that the nonlinearity is polynomial, and that the numerical tra jectory remains at least uni-…
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 study the long time behavior of small solutions of semi-linear dispersive Hamiltonian partial differential equations on confined domains. Provided that the system enjoys a new non-resonance condition and a strong enough energy estimate,…
These notes are based on lectures held at the Lanzhou university (China) during a CIMPA summer school in july 2004 but benefit from recent devellopements. Our aim is to explain some perturbations technics that allow to study the long time…
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…
A normal form is derived for Hamiltonian-Hopf bifurcations of solitary waves in generalized nonlinear Schr\"odinger equations. This normal form is a simple second-order nonlinear ordinary differential equation that is asymptotically…
This article gives a simple treatment of the quantum Birkhoff normal form for semiclassical pseudo-differential operators with smooth coefficients. The normal form is applied to describe the discrete spectrum in a generalised non-degenerate…
We consider the semilinear harmonic oscillator $$i\psi_t=(-\Delta +\va{x}^{2} +M)\psi +\partial_2 g(\psi,\bar \psi), \quad x\in \R^d, t\in \R$$ where $M$ is a Hermite multiplier and $g$ a smooth function globally of order 3 at least. We…
In this article, we study the semi discrete and fully discrete formulations for a Kirchhoff type quasilinear integro-differential equation involving time-fractional derivative of order $\alpha \in (0,1) $. For the semi discrete formulation…
Consider two kinds of 1-d Hamiltonian Derivative Nonlinear Schr\"odinger (DNLS) equations with respect to different symplectic forms under periodic boundary conditions. The nonlinearities of these equations depend not only on…
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…
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…
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 consider a general class of infinite dimensional reversible differential systems. Assuming a non resonance condition on the linear frequencies, we construct for such systems almost invariant pseudo norms that are closed to Sobolev-like…
We consider the non linear wave equation (NLW) on the d-dimensional torus with a smooth nonlinearity of order at least two at the origin. We prove that, for almost any mass, small and smooth solutions of high Sobolev indices are stable up…
This paper develops and analyzes a fully discrete finite element method for a class of semilinear stochastic partial differential equations (SPDEs) with multiplicative noise. The nonlinearity in the diffusion term of the SPDEs is assumed to…
In this article, we construct and analyse an explicit numerical splitting method for a class of semi-linear stochastic differential equations (SDEs) with additive noise, where the drift is allowed to grow polynomially and satisfies a global…
This paper is concerned with the derivative nonlinear Schr\"{o}dinger equation with periodic boundary conditions. We obtain complete Birkhoff normal form of order six. As an application, the long time stability for solutions of small…