30 research outputs found
Resonances in long time integration of semi linear Hamiltonian PDEs
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. The goal of this work is twofold: First we show how standard
approximation methods cannot in general avoid resonances issues, and we give
numerical examples of pathological behavior for the midpoint rule and
implicit-explicit integrators. Such phenomena can be avoided by suitable
truncations of the linear unbounded operator combined with classical splitting
methods. We then give a sharp bound for the cut-off depending on the time step.
Using a new normal form result, we show the long time preservation of the
actions for such schemes for all values of the time step, provided the initial
continuous system does not exhibit resonant frequencies
Exponentially accurate Hamiltonian embeddings of symplectic A-stable Runge--Kutta methods for Hamiltonian semilinear evolution equations
We prove that a class of A-stable symplectic Runge--Kutta time
semidiscretizations (including the Gauss--Legendre methods) applied to a class
of semilinear Hamiltonian PDEs which are well-posed on spaces of analytic
functions with analytic initial data can be embedded into a modified
Hamiltonian flow up to an exponentially small error. As a consequence, such
time-semidiscretizations conserve the modified Hamiltonian up to an
exponentially small error. The modified Hamiltonian is -close to the
original energy where is the order of the method and the time
step-size. Examples of such systems are the semilinear wave equation or the
nonlinear Schr\"odinger equation with analytic nonlinearity and periodic
boundary conditions. Standard Hamiltonian interpolation results do not apply
here because of the occurrence of unbounded operators in the construction of
the modified vector field. This loss of regularity in the construction can be
taken care of by projecting the PDE to a subspace where the operators occurring
in the evolution equation are bounded and by coupling the number of excited
modes as well as the number of terms in the expansion of the modified vector
field with the step size. This way we obtain exponential estimates of the form
with and ; for the semilinear wave
equation, , and for the nonlinear Schr\"odinger equation, . We give
an example which shows that analyticity of the initial data is necessary to
obtain exponential estimates
Stroboscopic Averaging for the Nonlinear Schrödinger Equation
International audienceIn this paper, we are concerned with an averaging procedure, - namely Stroboscopic averaging [SVM07, CMSS10] -, for highly-oscillatory evolution equations posed in a (possibly infinite dimensional) Banach space, typically partial differential equations (PDEs) in a high-frequency regime where only one frequency is present. We construct a highorder averaged system whose solution remains exponentially close to the exact one over long time intervals, possesses the same geometric properties (structure, invariants, . . . ) as compared to the original system, and is non-oscillatory. We then apply our results to the nonlinear Schrödinger equation on the d-dimensional torus , or in with a harmonic oscillator, for which we obtain a hierarchy of Hamiltonian averaged models. Our results are illustrated numerically on several examples borrowed from the recent literature
KAM theory for the Hamiltonian derivative wave equation
We prove an infinite dimensional KAM theorem which implies the existence of
Cantor families of small-amplitude, reducible, elliptic, analytic, invariant
tori of Hamiltonian derivative wave equationsComment: 66 page
Chaotic-like transfers of energy in Hamiltonian PDEs
We consider the nonlinear cubic Wave, the Hartree and the nonlinear cubic
Beam equations on and we prove the existence of different types of
solutions which exchange energy between Fourier modes in certain time scales.
This exchange can be considered \emph{chaotic-like} since either the choice of
activated modes or the time spent in each transfer can be chosen randomly. The
key point of the construction of those orbits is the existence of heteroclinic
connections between invariant objects and the construction of symbolic dynamics
(a Smale horseshoe) for the Birkhoff Normal Form truncation of those equations