# KAM theory and Geometric Integration (11w5132)

Arriving in Banff, Alberta Sunday, June 5 and departing Friday June 10, 2011

## Organizers

Walter Craig (McMaster University)

Erwan Faou (Ecole Normale Superieure de Cachan Bretagne)

Benoit Grébert (Universite de Nantes, France)

## Objectives

$ {\bf KAM theory and Geometric Integration} $

The principal equations of mathematical physics, whether it is

quantum mechanics, Bose -- Einstein condensates, molecular dynamics,

ocean waves, the $n$-body problem of celestial mechanics, or

Einstein's equations in general relativity, are in fact Hamiltonian

dynamical systems when viewed in the proper coordinates. This

basic observation lends weight to the mathematical point of view

that the topics of dynamical systems and nonlinear partial

differential equations should be considered to be cousins, and

have a lot in common. In particular this point of view has lead

to the consideration of the global behavior of orbits of a

Hamiltonian PDE in an appropriate phase space, the pursuit

of the mathematical technology of normal forms, the study

of stable orbits and KAM tori, possibly of infinite dimension,

and a number of results analogous to Nekhoroshev stability and Arnold

diffusion. Over the past decade there has been a number of important

contributions to this point of view from a theoretical standpoint.

On the other hand, it is a very important direction of research to

adapt the point of view of Hamiltonian dynamics to numerical

simulations of (at least some of) the physical phenomena mentioned

above. Some major past achievements include developments of symplectic

numerical integrator routines, and their use in fluid mechanics and

large scale partical dynamics computations. The focus of this

workshop at BIRS is to bring the Hamiltonian PDE and the scientific

computing communities together, to reflect on future common directions

of research, to encourage the development of numerical methods to

effectively model the evolution of continuum systems possessing

infinitely many degrees of freedom, and to communicate the most

up-to-date theoretical and numerical research.

$ {\it Geometric Integration:} $

One of the major goal of geometric integration is to analyze the

qualitative behavior of numerical methods applied to Hamiltonian

Systems over long time. The principal motivation comes from the

fact that in many applications, the preservation of the geometrical

structure of the flow is more important than the approximation of

the trajectory or a particle itself. This is for example typically

the case in molecular dynamics, where the energy preservation over

long time and the conservation of the volume allows to use the

numerical solution as a sampling method to explore the phase

space by preserving a given invariant measure. In many other

situations, the preservation of physical invariants (momentum,

energy) is crucial, and does not hold automatically, even for

numerical scheme with a high approximation degree. Geometric

integration can hence be understood as the study of geometric

numerical methods (e.g. symplectic for Hamiltonian systems)

and the possible replication of geometrical properties of the

exact flow by these methods.

For finite dimensional Hamiltonian systems, the situation is

now well understood: The results given by the {em backward

error analysis} state that the discrete numerical solution

can be be interpreted as the exact solution at discrete time

of a perturbed differential systems (the perturbation parameter

being here the step size), up to exponentially small error terms (see $ \cite{HLW,Reic04}). $

Moreover, in the case of Hamiltonian systems, and if the numerical

integrator is symplectic, the perturbed differential systems carries

on the symplectic strucuture, and hence the perturbation can be seen

as a small perturbation of the Hamiltonian.

These result have major outcomes in many applications fields.

Recently, Hairer, Lubich and Wanner used theses result, and

combined them with classical techniques developed in KAM theory

to show that in the numerical simulation of completely integrable

systems by symplectic integrators, KAM tori are preserved under

non resonances conditions on the initial systems (or non resonance

conditions over the step size). Moreover, all the classical results

of perturbation theory can be applied to this numerical setting: In

particular Nekhoroshev estimates can be derived, or existence of

dense sets of numerical KAM tori can be shown.

These kind of results have consequences in many applications, from

molecular dynamics to celestial mechanics.

The central result given by the backward error analysis is valid

under the stability hypothesis that the trajectory remains in a

compact set of the phase space, and that the step size is small

with respect to the inverse of the highest eigenvalue of the

system. These two assumptions make that it is impossible to

use directly these techniques for the understanding of the

numerical simulation of partial differential equations over

long time, where the discretized systems possess eigenvalues

of arbitrary size.

Because of this latter fact, most of the standard symplectic numerical schemes applied to Hamiltonian PDEs cannot avoid the presence of {em numerical resonances}: for some specific value of the stepsize depending on the frequencies of the original system, the preservation properties (energy, invariants..) are lost by the numerical solution.

Finding new mathematical tools to replace the

use of backward error analysis in the case of PDEs or highly

oscillatory systems is today a formidable challenge in geometrical

integration.

The fact that backward error analysis does not apply to partial

differential equations leads to technical innovations in the field.

Recently, significant progresses have been made in this direction:

Hairer and Lubich invented the {em modulated Fourier expansion}

aiming at representing the solutions of the a Hamiltonian partial

differential equations in term of functions oscillating at the

frequencies of the linear underlying operator (see $ \cite{CHL08a,CHL08c,CHL08b,GL08a,GL08b}. $Strikingly, they

recover results given by Bambusi and Gr'ebert (see $ \cite{Bam03,Greb07}) $ concerning the

long-time behavior of the non linear wave equation and Schr"odinger

equations, and derived similar results for their numerical approximations. Another technology employed to replace backward error

analysis comes from the use of {em normal form techniques}, and

was applied by Dujardin and Faou to the numerical approximation

of the linear Schr"odinger equation by splitting methods (see $ \cite{DF07}). $

These normal form techniques were adapted to the case of semilinear Hamiltonian PDEs by Faou, Gr'ebert and Paturel and their approximations by symplectic splitting methods (see $ \cite{FGP1,FGP2}). $

$ {\it Hamiltonian PDEs:} $

It turns out that while these questions came to maturity from

the numerical analysis point of view, many

progresses were made concerning the theoretical study of non

linear Hamiltonian PDE over long time:

KAM like results were obtained mainly by Kuksin, Craig, Eliasson,

Wayne, Bourgain, P"oschel, normal form results by Bourgain, Bambusi,

Gr'ebert (see for instance cite{ElKuk,[46],KP,K1,K2,Poe,CW93,Bou96,BG06}). The possible extensions of such results is a fundamental

question, in particular in situations corresponding to real cases

of applications such as plasma physics and fluid simulations.

Actually, normal form theory for nonlinear PDEs has reached a satisfactory

development for what concerns the study of semilinear equations in

space dimension 1. In higher space dimensions there are only a few

examples that at present we are able to treat. The development of a

satisfactory theory requires some new ideas, due to the greater

complexity of small denominators in arbitrary dimension. A second

important problem concerns the case where the nonlinearity involves

derivatives. A comprehension of this situation would be of major

importance since most of the models coming from physics are of this

kind.

Hence many questions arise simultaneously both from the theoretical

and practical points of view. Another example is given by limiting

questions induced by the studying of large Hamiltonian lattices:

Typically, Fermi-Pasta-Ulam systems converge in the limit of a

large number of particle towards Hamiltonian partial differential

equation (KdV), or alternatively can be viewed as discretizations

of these PDEs. Hence, all the tools used to analyze these systems

(normal forms techniques independent of the dimension, modulated

Fourier expansion independent of the space discretization) can

be interpreted as tentative explanations of the same phenomenon.

$ {\bf Objectives and novelty of the workshop} $

The main goal of this workshop is to bring together people working

on the qualitative behavior of the solutions of Hamiltonian systems

(ordinary differential equations or partial differential equation)

both from the theoretical and numerical points of view.

By essence, the techniques used in these two fields are very close:

high frequency cut-off, control of the regularity, error estimates,

etc. We believe that the mixing of people coming from these two

communities can be a benefit in many directions:

$

\begin{itemize}

\item It can bring new problems typically arising from numerical

issues. For instance, the role of backward error analysis is to

make the link between an information on the flow (what is

implemented in the computer is a perturbation of the exact flow)

to an information that can be read on the vector field. From the

practical point of view, this produces new and purely numerical

resonances effects that has to be understood, if not avoided by

the use of new adapted numerical schemes.

\item The analysis of the long-time behavior of numerical schemes applied to nonlinear PDEs is an emerging field. The development and analysis of efficient numerical methods in this context require technical innovations.

The interaction of these people could allow to develop

new tools in both directions: Can modulated Fourier expansions

be used to prove new theoretical results? Can normal form-like

techniques allow to prove new results from the numerical point

of view? Can analytical tools be used to understand existing

numerical schemes used in many applications? Can the same techniques

be used to invent new efficients numerical schemes?

\item Another benefit would be the extension to other type of

equations coming directly from the application: Simulation of

Plasma and Vlasov-Maxwell equations, numerical simulations of free

surface water waves using the Hamiltonian structure of Zakharov

and the spectral implementation of the Dirichlet -- Neumann operator

of Craig, Guyenne $&$ Sulem, further simulation of fluid dynamics in an

inviscid regime. The use of symplectic integrator and the

understanding of their behavior lead to many unresolved questions,

both the theoretical and numerical point of view.

\end{itemize} $

$

\begin{thebibliography}{30}

\bibitem{Bam03}

D.~Bambusi, emph{Birkhoff normal form for some nonlinear {PDE}s}, Comm. Math.

Physics 234 (2003), 253--283.

\bibitem{BG06}

{rm D. Bambusi and B. Gr'ebert},

{em Birkhoff normal form for PDE's with tame modulus}. Duke Math. J. 135 no. 3 (2006), 507ï¿½-567.

\bibitem{Bou96}

{rm J. Bourgain},

{em Construction of approximative and almost-periodic solutions of perturbed linear Schr"odinger and wave equations}, Geometric

and functional Analysis, 6, (1996) 201--230.

\bibitem{CHL08a}

{rm D. Cohen, E. Hairer and C. Lubich},

{em Long-time analysis of nonlinearly perturbed wave equations via modulated Fourier expansions},

Arch. Ration. Mech. Anal. 187 (2008) 341-368.

\bibitem{CHL08c}

{rm D. Cohen, E. Hairer and C. Lubich},

{em Conservation of energy, momentum and actions in numerical discretizations of nonlinear wave equations},

Numerische Mathematik 110 (2008) 113--143.

\bibitem{[75]} {it Hamiltonian dynamical systems and applications}

(W. Craig, editor),

Proceedings of the Advanced Study Institute on Hamiltonian

Dynamical Systems and Applications,

NATO Science for Peace and Security Series B: Springer - Verlag,

(2008) XVI, 441~pp.

\bibitem{[46]} {rm W. Craig}, ``Probl`emes de petits diviseurs dans les 'equations aux

d'eriv'ees partielles'', {sl Panaromas et Synth`eses bf 9}, Soci'et'e

Math'ematiques de France (2000).

\bibitem{CW93}

{rm W. Craig and C. E. Wayne}

{em Newton's method and periodic solutions of nonlinear wave equations}, Comm. Pure Appl. Math. 46 (1993), 1409--1498.

\bibitem{DF07}

{rm G. Dujardin and E. Faou},

{em Normal form and long time analysis of splitting schemes for the linear Schr{"o}dinger equation with small potential.}

Numer. Math. 106, 2 (2007) 223--262

\bibitem{ElKuk}

{rm L. H. Eliasson and S. B. Kuksin},

{em KAM for the non-linear Schr"odinger equation}, to appear in Annals of Math. (2009)

\bibitem{FGP1}

{rm E. Faou, B. Gr'ebert and E. Paturel},

{em Birkhoff normal form for splitting methods applied to semi linear Hamiltonian PDEs. Part I: Finite dimensional discretization.} To appear in Numer. Math. (2009).

\bibitem{FGP2}

{rm E. Faou, B. Gr'ebert and E. Paturel},

{em Birkhoff normal form for splitting methods applied to semi linear Hamiltonian PDEs. Part II: Abstract splitting.} To appear in Numer. Math. (2009).

\bibitem{GL08a}

{rm L. Gauckler and C. Lubich},

{em Nonlinear Schr"odinger equations and their spectral discretizations over long times},

To appear in Found. Comput. Math. (2009).

\bibitem{GL08b}

{rm L. Gauckler and C. Lubich},

{em Splitting integrators for nonlinear Schr"odinger equations over long times},

To appear in Found. Comput. Math. (2009).

\bibitem{Greb07}

{rm B. Gr'ebert},

{em Birkhoff normal form and Hamiltonian PDEs}, Partial differential equations and applications,

S'emin. Congr.,

vol.~15, Soc. Math. France, Paris, 2007, pp.~1--46.

\bibitem{CHL08b}

{rm E. Hairer and C. Lubich},

{em Spectral semi-discretisations of weakly nonlinear wave equations over long times},

Found. Comput. Math. 8 (2008) 319-334.

\bibitem{HLW}

{rm E. Hairer, C. Lubich and G. Wanner},

{em Geometric Numerical Integration. Structure-Preserving Algorithms for Ordinary Differential Equations}. Second Edition. Springer 2006.

\bibitem{KP}

{rm T. Kappeler and J. P"oschel},

{em KdV $&$ KAM.}

rm Springer, Berlin, 2003.

\bibitem{K1}

{rm S. Kuksin}. emph{Nearly integrable infinite-dimensional {H}amiltonian systems}, Springer-Verlag, Berlin, 1993.

\bibitem{K2}

{rm S. Kuksin} emph{Analysis of {H}amiltonian {PDEs}}, Oxford University Press, Oxford, 2000.

\bibitem{Reic04}

{rm B. Leimkuhler, S. Reich},

{em Simulating Hamiltonian dynamics}.

Cambridge Monographs on Applied and Computational Mathematics, 14. Cambridge University Press, Cambridge, 2004.

\bibitem{Poe}

{rm J. P"oschel}, {em A KAM-theorem for some nonlinear PDEs}, Ann. Scuola Norm.

Sup. Pisa, Cl. Sci., IV Ser. 15 23 (1996), 119--148.

\end{thebibliography}

$

The principal equations of mathematical physics, whether it is

quantum mechanics, Bose -- Einstein condensates, molecular dynamics,

ocean waves, the $n$-body problem of celestial mechanics, or

Einstein's equations in general relativity, are in fact Hamiltonian

dynamical systems when viewed in the proper coordinates. This

basic observation lends weight to the mathematical point of view

that the topics of dynamical systems and nonlinear partial

differential equations should be considered to be cousins, and

have a lot in common. In particular this point of view has lead

to the consideration of the global behavior of orbits of a

Hamiltonian PDE in an appropriate phase space, the pursuit

of the mathematical technology of normal forms, the study

of stable orbits and KAM tori, possibly of infinite dimension,

and a number of results analogous to Nekhoroshev stability and Arnold

diffusion. Over the past decade there has been a number of important

contributions to this point of view from a theoretical standpoint.

On the other hand, it is a very important direction of research to

adapt the point of view of Hamiltonian dynamics to numerical

simulations of (at least some of) the physical phenomena mentioned

above. Some major past achievements include developments of symplectic

numerical integrator routines, and their use in fluid mechanics and

large scale partical dynamics computations. The focus of this

workshop at BIRS is to bring the Hamiltonian PDE and the scientific

computing communities together, to reflect on future common directions

of research, to encourage the development of numerical methods to

effectively model the evolution of continuum systems possessing

infinitely many degrees of freedom, and to communicate the most

up-to-date theoretical and numerical research.

$ {\it Geometric Integration:} $

One of the major goal of geometric integration is to analyze the

qualitative behavior of numerical methods applied to Hamiltonian

Systems over long time. The principal motivation comes from the

fact that in many applications, the preservation of the geometrical

structure of the flow is more important than the approximation of

the trajectory or a particle itself. This is for example typically

the case in molecular dynamics, where the energy preservation over

long time and the conservation of the volume allows to use the

numerical solution as a sampling method to explore the phase

space by preserving a given invariant measure. In many other

situations, the preservation of physical invariants (momentum,

energy) is crucial, and does not hold automatically, even for

numerical scheme with a high approximation degree. Geometric

integration can hence be understood as the study of geometric

numerical methods (e.g. symplectic for Hamiltonian systems)

and the possible replication of geometrical properties of the

exact flow by these methods.

For finite dimensional Hamiltonian systems, the situation is

now well understood: The results given by the {em backward

error analysis} state that the discrete numerical solution

can be be interpreted as the exact solution at discrete time

of a perturbed differential systems (the perturbation parameter

being here the step size), up to exponentially small error terms (see $ \cite{HLW,Reic04}). $

Moreover, in the case of Hamiltonian systems, and if the numerical

integrator is symplectic, the perturbed differential systems carries

on the symplectic strucuture, and hence the perturbation can be seen

as a small perturbation of the Hamiltonian.

These result have major outcomes in many applications fields.

Recently, Hairer, Lubich and Wanner used theses result, and

combined them with classical techniques developed in KAM theory

to show that in the numerical simulation of completely integrable

systems by symplectic integrators, KAM tori are preserved under

non resonances conditions on the initial systems (or non resonance

conditions over the step size). Moreover, all the classical results

of perturbation theory can be applied to this numerical setting: In

particular Nekhoroshev estimates can be derived, or existence of

dense sets of numerical KAM tori can be shown.

These kind of results have consequences in many applications, from

molecular dynamics to celestial mechanics.

The central result given by the backward error analysis is valid

under the stability hypothesis that the trajectory remains in a

compact set of the phase space, and that the step size is small

with respect to the inverse of the highest eigenvalue of the

system. These two assumptions make that it is impossible to

use directly these techniques for the understanding of the

numerical simulation of partial differential equations over

long time, where the discretized systems possess eigenvalues

of arbitrary size.

Because of this latter fact, most of the standard symplectic numerical schemes applied to Hamiltonian PDEs cannot avoid the presence of {em numerical resonances}: for some specific value of the stepsize depending on the frequencies of the original system, the preservation properties (energy, invariants..) are lost by the numerical solution.

Finding new mathematical tools to replace the

use of backward error analysis in the case of PDEs or highly

oscillatory systems is today a formidable challenge in geometrical

integration.

The fact that backward error analysis does not apply to partial

differential equations leads to technical innovations in the field.

Recently, significant progresses have been made in this direction:

Hairer and Lubich invented the {em modulated Fourier expansion}

aiming at representing the solutions of the a Hamiltonian partial

differential equations in term of functions oscillating at the

frequencies of the linear underlying operator (see $ \cite{CHL08a,CHL08c,CHL08b,GL08a,GL08b}. $Strikingly, they

recover results given by Bambusi and Gr'ebert (see $ \cite{Bam03,Greb07}) $ concerning the

long-time behavior of the non linear wave equation and Schr"odinger

equations, and derived similar results for their numerical approximations. Another technology employed to replace backward error

analysis comes from the use of {em normal form techniques}, and

was applied by Dujardin and Faou to the numerical approximation

of the linear Schr"odinger equation by splitting methods (see $ \cite{DF07}). $

These normal form techniques were adapted to the case of semilinear Hamiltonian PDEs by Faou, Gr'ebert and Paturel and their approximations by symplectic splitting methods (see $ \cite{FGP1,FGP2}). $

$ {\it Hamiltonian PDEs:} $

It turns out that while these questions came to maturity from

the numerical analysis point of view, many

progresses were made concerning the theoretical study of non

linear Hamiltonian PDE over long time:

KAM like results were obtained mainly by Kuksin, Craig, Eliasson,

Wayne, Bourgain, P"oschel, normal form results by Bourgain, Bambusi,

Gr'ebert (see for instance cite{ElKuk,[46],KP,K1,K2,Poe,CW93,Bou96,BG06}). The possible extensions of such results is a fundamental

question, in particular in situations corresponding to real cases

of applications such as plasma physics and fluid simulations.

Actually, normal form theory for nonlinear PDEs has reached a satisfactory

development for what concerns the study of semilinear equations in

space dimension 1. In higher space dimensions there are only a few

examples that at present we are able to treat. The development of a

satisfactory theory requires some new ideas, due to the greater

complexity of small denominators in arbitrary dimension. A second

important problem concerns the case where the nonlinearity involves

derivatives. A comprehension of this situation would be of major

importance since most of the models coming from physics are of this

kind.

Hence many questions arise simultaneously both from the theoretical

and practical points of view. Another example is given by limiting

questions induced by the studying of large Hamiltonian lattices:

Typically, Fermi-Pasta-Ulam systems converge in the limit of a

large number of particle towards Hamiltonian partial differential

equation (KdV), or alternatively can be viewed as discretizations

of these PDEs. Hence, all the tools used to analyze these systems

(normal forms techniques independent of the dimension, modulated

Fourier expansion independent of the space discretization) can

be interpreted as tentative explanations of the same phenomenon.

$ {\bf Objectives and novelty of the workshop} $

The main goal of this workshop is to bring together people working

on the qualitative behavior of the solutions of Hamiltonian systems

(ordinary differential equations or partial differential equation)

both from the theoretical and numerical points of view.

By essence, the techniques used in these two fields are very close:

high frequency cut-off, control of the regularity, error estimates,

etc. We believe that the mixing of people coming from these two

communities can be a benefit in many directions:

$

\begin{itemize}

\item It can bring new problems typically arising from numerical

issues. For instance, the role of backward error analysis is to

make the link between an information on the flow (what is

implemented in the computer is a perturbation of the exact flow)

to an information that can be read on the vector field. From the

practical point of view, this produces new and purely numerical

resonances effects that has to be understood, if not avoided by

the use of new adapted numerical schemes.

\item The analysis of the long-time behavior of numerical schemes applied to nonlinear PDEs is an emerging field. The development and analysis of efficient numerical methods in this context require technical innovations.

The interaction of these people could allow to develop

new tools in both directions: Can modulated Fourier expansions

be used to prove new theoretical results? Can normal form-like

techniques allow to prove new results from the numerical point

of view? Can analytical tools be used to understand existing

numerical schemes used in many applications? Can the same techniques

be used to invent new efficients numerical schemes?

\item Another benefit would be the extension to other type of

equations coming directly from the application: Simulation of

Plasma and Vlasov-Maxwell equations, numerical simulations of free

surface water waves using the Hamiltonian structure of Zakharov

and the spectral implementation of the Dirichlet -- Neumann operator

of Craig, Guyenne $&$ Sulem, further simulation of fluid dynamics in an

inviscid regime. The use of symplectic integrator and the

understanding of their behavior lead to many unresolved questions,

both the theoretical and numerical point of view.

\end{itemize} $

$

\begin{thebibliography}{30}

\bibitem{Bam03}

D.~Bambusi, emph{Birkhoff normal form for some nonlinear {PDE}s}, Comm. Math.

Physics 234 (2003), 253--283.

\bibitem{BG06}

{rm D. Bambusi and B. Gr'ebert},

{em Birkhoff normal form for PDE's with tame modulus}. Duke Math. J. 135 no. 3 (2006), 507ï¿½-567.

\bibitem{Bou96}

{rm J. Bourgain},

{em Construction of approximative and almost-periodic solutions of perturbed linear Schr"odinger and wave equations}, Geometric

and functional Analysis, 6, (1996) 201--230.

\bibitem{CHL08a}

{rm D. Cohen, E. Hairer and C. Lubich},

{em Long-time analysis of nonlinearly perturbed wave equations via modulated Fourier expansions},

Arch. Ration. Mech. Anal. 187 (2008) 341-368.

\bibitem{CHL08c}

{rm D. Cohen, E. Hairer and C. Lubich},

{em Conservation of energy, momentum and actions in numerical discretizations of nonlinear wave equations},

Numerische Mathematik 110 (2008) 113--143.

\bibitem{[75]} {it Hamiltonian dynamical systems and applications}

(W. Craig, editor),

Proceedings of the Advanced Study Institute on Hamiltonian

Dynamical Systems and Applications,

NATO Science for Peace and Security Series B: Springer - Verlag,

(2008) XVI, 441~pp.

\bibitem{[46]} {rm W. Craig}, ``Probl`emes de petits diviseurs dans les 'equations aux

d'eriv'ees partielles'', {sl Panaromas et Synth`eses bf 9}, Soci'et'e

Math'ematiques de France (2000).

\bibitem{CW93}

{rm W. Craig and C. E. Wayne}

{em Newton's method and periodic solutions of nonlinear wave equations}, Comm. Pure Appl. Math. 46 (1993), 1409--1498.

\bibitem{DF07}

{rm G. Dujardin and E. Faou},

{em Normal form and long time analysis of splitting schemes for the linear Schr{"o}dinger equation with small potential.}

Numer. Math. 106, 2 (2007) 223--262

\bibitem{ElKuk}

{rm L. H. Eliasson and S. B. Kuksin},

{em KAM for the non-linear Schr"odinger equation}, to appear in Annals of Math. (2009)

\bibitem{FGP1}

{rm E. Faou, B. Gr'ebert and E. Paturel},

{em Birkhoff normal form for splitting methods applied to semi linear Hamiltonian PDEs. Part I: Finite dimensional discretization.} To appear in Numer. Math. (2009).

\bibitem{FGP2}

{rm E. Faou, B. Gr'ebert and E. Paturel},

{em Birkhoff normal form for splitting methods applied to semi linear Hamiltonian PDEs. Part II: Abstract splitting.} To appear in Numer. Math. (2009).

\bibitem{GL08a}

{rm L. Gauckler and C. Lubich},

{em Nonlinear Schr"odinger equations and their spectral discretizations over long times},

To appear in Found. Comput. Math. (2009).

\bibitem{GL08b}

{rm L. Gauckler and C. Lubich},

{em Splitting integrators for nonlinear Schr"odinger equations over long times},

To appear in Found. Comput. Math. (2009).

\bibitem{Greb07}

{rm B. Gr'ebert},

{em Birkhoff normal form and Hamiltonian PDEs}, Partial differential equations and applications,

S'emin. Congr.,

vol.~15, Soc. Math. France, Paris, 2007, pp.~1--46.

\bibitem{CHL08b}

{rm E. Hairer and C. Lubich},

{em Spectral semi-discretisations of weakly nonlinear wave equations over long times},

Found. Comput. Math. 8 (2008) 319-334.

\bibitem{HLW}

{rm E. Hairer, C. Lubich and G. Wanner},

{em Geometric Numerical Integration. Structure-Preserving Algorithms for Ordinary Differential Equations}. Second Edition. Springer 2006.

\bibitem{KP}

{rm T. Kappeler and J. P"oschel},

{em KdV $&$ KAM.}

rm Springer, Berlin, 2003.

\bibitem{K1}

{rm S. Kuksin}. emph{Nearly integrable infinite-dimensional {H}amiltonian systems}, Springer-Verlag, Berlin, 1993.

\bibitem{K2}

{rm S. Kuksin} emph{Analysis of {H}amiltonian {PDEs}}, Oxford University Press, Oxford, 2000.

\bibitem{Reic04}

{rm B. Leimkuhler, S. Reich},

{em Simulating Hamiltonian dynamics}.

Cambridge Monographs on Applied and Computational Mathematics, 14. Cambridge University Press, Cambridge, 2004.

\bibitem{Poe}

{rm J. P"oschel}, {em A KAM-theorem for some nonlinear PDEs}, Ann. Scuola Norm.

Sup. Pisa, Cl. Sci., IV Ser. 15 23 (1996), 119--148.

\end{thebibliography}

$