# KAM theory and Geometric Integration (11w5132)

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

## Organizers

(McMaster University)

(Ecole Normale Superieure de Cachan Bretagne)

(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}, Problemes de petits diviseurs dans les 'equations aux d'eriv'ees partielles'', {sl Panaromas et Syntheses 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$&$KAM.} rm Springer, Berlin, 2003. \bibitem{K1} {rm S. Kuksin}. emph{Nearly integrable infinite-dimensional {H}amiltonian systems}, Springer-Verlag, Berlin, 1993. \bibitem{K2} {rm S. Kuksin} emph{Analysis of {H}amiltonian {PDEs}}, Oxford University Press, Oxford, 2000. \bibitem{Reic04} {rm B. Leimkuhler, S. Reich}, {em Simulating Hamiltonian dynamics}. Cambridge Monographs on Applied and Computational Mathematics, 14. Cambridge University Press, Cambridge, 2004. \bibitem{Poe} {rm J. P"oschel}, {em A KAM-theorem for some nonlinear PDEs}, Ann. Scuola Norm. Sup. Pisa, Cl. Sci., IV Ser. 15 23 (1996), 119--148. \end{thebibliography}$