Indam intensive period New connections between dynamical systems and Hamiltonian PDEs NAPOLI, April 1- June 6, 2009

Speakers and Titles

first period:introduction

• Pietro Baldi (Università of Napoli "Federico II", Napoli, Italy): An introduction to water waves (first period)
• Luca Biasco (Università "Roma Tre", Roma, Italy): An Introduction to KAM theory (10 hours, first week of April)
• Philippe Bolle (Université de Avignon) (April, 19-26)
• Piero D'Ancona (Università di Roma "La Sapienza", Roma, Italy): Introduction to dispersive equations. (18 hours, April 27 to May, 15)
• Amadeu Delshams (Universitat Politècnica de Catalunya, Barcelona, Spain), Instability in Hamiltonian systems via geometrical methods (2 lectures, April 15-18)
• Ivar Ekeland, (April 20-May 2)
• Donato Fortunato (Università of Bari, Italy) "Solitary Waves, Vortices and Electromagnetic Fields" (April 20-24)
• Guido Gentile (Università "Roma Tre", Roma, Italy) (April 15)
• Gérard Iooss, (I.U.F., Université de Nice) An example of infinite dimensional reversible system: Water wave theory, (3 lectures, May 4-9)
• Alessandro Portaluri (Università del Salento, Lecce, Italy): Introduction to symplectic invariants. (8 hours, May 11-15)
• Michela Procesi (Università of Napoli "Federico II", Napoli, Italy) (first period)
• Alfonso Sorrentino (Paris IX, Paris, France), An introduction to Mather's theory, (8 hours, April 6-17)
• Fabio Zanolin (Università of Udine, Udine, Italy), Some examples of "slow chaos" in simple planar Hamiltonian systems. (1 lecture, April 13-17)

second period: advanced school

• James Colliander (University of Toronto, Toronto, Canada): Maximal-in-time dynamics for nonlinear Schrödinger equations (5 lectures, May 25-29)
• Norman Dancer (University of Sydney, Sidney, Australia): Local and global bifurcation for nonlinear mappings and applications (5 lectures, May 25-29)
• Lars-Hakan Eliasson (Paris VII, Paris, France): Quasi-periodic Schrödinger operators (5 lectures, May 18-22)
• Patrick Gerard (Paris-Sud, Paris, France): The Cauchy Problem for Weakly Dispersive Nonlinear Schrödinger Equations, (5 lectures, May 25-29)
• Thomas Kappeler (ETH Zurich, Switzerland), Jürgen Pöschel (University of Stuttgart, Germany): Normal form theory of the defocusing nonlinear Schrödinger equation, (5 lectures, May 18-22)
• Sergei B. Kuksin (École Polytechnique, Palaiseau, France): KAM for PDEs (5 lectures, May 25-29)
• Eduard J. Zehnder (ETH Zurich): Periodic orbits of Hamiltonian systems and symplectic invariants, (5 lectures, May 18-22)
• Evgeny Korotyaev (Cardiff Univ., UK): Hamiltonian for KDV as a function of Marchenko-Ostrovki parameters (1 talk)

Workshop

• D. Bambusi
• V. Benci
• F. Calogero
• Cheng Chong-Qing
• L. Chierchia
• J. Colliander
• W. Craig
• D. Dolgopyat
• R. de la Llave
• G. Gallavotti
• J. Geng
• V. Georgiev
• B. Grébert
• M. Groves
• V. Y. Kaloshin
• A. Kiselev
• L. Stolovich
• S. Terracini
• J. Toland
• C. Viterbo
• S. Volberg
• W.M. Wang
• X. Yuan
• J. You

Abstracts

1. Zakharov formulation of Euler's equations of hydrodynamics via conformal mappings
2. Variational principles and Hamiltonian structure

1. the classical Kolmogorov-Arnold-Moser theorem for lagrangian tori of finite dimensional Hamiltonian systems
2. the KAM theory for lower dimensional elliptic tori
3. the KAM theory for infinite dimensional Hamiltonian systems and applications to 1-dim. PDEs

References (can be downloaded from the web site of Jürgen Pöschel)

1. Pöschel, Jürgen. A lecture on the classical KAM theorem. Smooth ergodic theory and its applications (Seattle, WA, 1999), 707-732, Proc. Sympos. Pure Math., 69, Amer. Math. Soc., Providence, RI, 2001.
2. Pöschel, Jürgen. On elliptic lower-dimensional tori in Hamiltonian systems. Math. Z. 202 (1989), no. 4, 559-608.
3. Pöschel, Jürgen. A KAM-theorem for some nonlinear partial differential equations. Ann. Scuola Norm. Sup. Pisa Cl. Sci. (4) 23 (1996), no. 1, 119--148.

1. Maximal-in-time issues for NLS
2. Global well-posedness below the energy norm
3. Low regularity theory for focusing NLS
4. I-method with a Morawetz bootstrap
5. Weakly turbulent solutions of cubic NLS on the 2d torus

I will give a self-contained introduction to modern techniques in the study of evolution equations of dispersive type; the main examples being the Schroedinger, wave, Klein-Gordon, Dirac and KdV equations. I will touch the following topics:

1. Basic tools (interpolation, function spaces, Paley-Littlewood dyadic techniques)
2. Dispersive and decay estimates
3. Smoothing estimates
4. Strichartz estimates
5. Applications to NLS, NLW: local and global existence, regularity issues

The basic global bifurcation theorem, the effect of symmetries, global bifurcation theorems where there are both symmetries and variational structure, applications to periodic solutions.

I will describe two different settings where the geometrical methods can be applied to detect (global) instability: a priori chaotic and a priori unstable Hamiltonian systems. A very wide class of geodesic flows in any dimension plus a quasi-periodic perturbation give rise to a priori chaotic Hamiltonian system, whereas a priori unstable Hamiltonian systems take place in considering periodic perturbations of a (or some) pendulum plus a (or some) rotor.

In both cases, there is a very big invariant object called *NHIM* (normally hyperbolic invariant manifold), which apart from its inner dynamics, possesses an outer dynamics, due to the transversal intersection of its associated unstable and stable invariant manifolds, which is described by the so called *scattering map*.

The combination of both dynamics along the *NHIM* gives rise to chaotic and unstable global behavior.

References (available from http://www.ma1.upc.edu/~amadeu/):

• Geography of resonances and Arnold diffusion in a priori unstable Hamiltonian systems; Amadeu Delshams, Gemma Huguet, 2008 (preprint).
• Geometric properties of the scattering map of a normally hyperbolic invariant manifold; Amadeu Delshams, Rafael de la Llave, Tere M. Seara, Adv. Math., 217(3):1096-1153, 2008.
• Orbits of unbounded energy in quasi-periodic perturbations of geodesic flows; Amadeu Delshams, Rafael de la Llave, Tere M. Seara, Adv. Math., 202(1):64-188, 2006.
• A Geometric Mechanism for Diffusion in Hamiltonian Systems Overcoming the large Gap Problem: Heuristics and Rigorous Verification on a Model; Amadeu Delshams, Rafael de la Llave, Tere M. Seara, Mem. Amer. Math. Soc., 179(844):1-141, 2006.

The lecture will treat the one-dimensional quasi-periodic Schrödinger operator and its relation to the Schrödinger co-cycle and the Schrödinger equation.

We shall present the rotation number and the Lyapunov exponent, and we shall shortly discuss their relations to spectral and dynamical properties [3]). This by now almost classical subject is well covered in the books [5] and [2].

The major part of the lectures will be devoted to the perturbation theory of this operator [4] in the two regimes:

• weak coupling where properties like (almost) reducibility and transport are predominant and
• strong coupling where point spectrum and localization are the dominant features.

If time permits we will discuss the relation of the perturbative results with the semi-perturbative results obtained recently in, for example, [1].

References

1. J. BOURGAIN, Green's function estimates for lattice Schrödinger operators and applications, Annals of Mathematical Studies, Princeton University Press, Princeton (2004).
2. R. CARMONA, J. LACROIX, Spectral theory of random Schrödinger operators, Birkhäuser, Boston (1990).
3. L.H. ELIASSON, One-dimensional quasi-periodic Schrödinger operators – Dynamical systems and spectral theory, in A. Balog et. al. (eds.): Proceedings of the European Congress of Mathematics, Budapest 1996, Birkhäuser Verlag, Basel, 1998.
4. L.H. ELIASSON, Perturbations of linear quasi-periodic systems, in S. Marmi, J.-C. Yoccoz (eds.) Dynamical Systems and Small divisors, Cetraro 1998, Springer, Berlin, 2002, 60 pages.
5. L. PASTUR, A. FIGOTIN, Spectra of random and almost-periodic operators, Grundlehren der mathematischen wissenschaften 297, Springer-Verlag, Berlin Heidelberg (1992).

Existence of smooth flow maps and bilinear Strichartz estimates on manifolds

1. The necessity of bilinear Strichartz estimates
2. Bourgain's method and the sufficiency of bilinear Strichartz estimates
3. Proof of bilinear Strichartz estimates on general manifolds and improvements on spheres and tori

The cubic Szegö equation: an integrable toy model for completely non dispersive Hamiltonian evolutions

1. Motivation. Cauchy problem. Lax pair
2. Large time behavior. Traveling waves

The introduction of spatial dynamics (K.Kirchgässner in the eighties) allowed big progresses in the mathematical theory of water waves. Several new forms of localized waves were discovered. The two first talks will give elements of the reduction methods (center manifold reduction and normal forms for infinite dimensional reversible systems) and examples of results for travelling waves in 2D and 3D water vave theory. For the 3D problem, the limiting case with no surface tension introduces a small divisor problem and the above reduction methods fail. This situation will be the subject of the last lecture which will present the method and results on travelling gravity waves with a horizontally periodic pattern.

Reference

1. F.Dias, G.Iooss. Water-waves as a spatial dynamical system. Handbook of math. fluid dynamics. Chap 10, p.443-499.
2. S.Friedlander, D.Serre Eds., Elsevier 2003. G.Iooss, P.Plotnikov. Small divisor problem in the theory of three-dimensional water gravity waves. to appear in Memoirs of AMS.

Our lectures are based on a book in preparation. Parts of the manuscript will be distributed to the students attending the lectures.

Lecture 1: Introduction
In the first lecture we will review the notion of integrable systems of finite dimension and their normal forms.

Lecture 2: Nonlinear Schrödinger equation (NLS)
In the second lecture we explain how the nonlinear Schrödinger equation on the circle can be viewed as a Hamiltonian system and show that it can be written as a Lax pair, involving the so called Zhakarov Shabat operator. For the defocusing NLS, this operator is self-adjoint. Its properties, in particular its spectrum when considered with periodic or Dirichlet boundary conditions will be discussed.

Lecture 3 and 4: Birkhoff normal form of NLS
In the third and fourth lecture we will present a quite general scheme how to construct Birkhoff coordinates for the defocusing NLS on the circle. They turn out to be globally defined.

Lecture 5: Applications
In the last lecture we will discuss applications of the Birkhoff coordinates to the well-posedness problem of the defocusing NLS on the circle in weighted Sobolev spaces. If time permits we will also discuss a KAM theorem for NLS.

We consider the KdV equation with periodic initial conditions. We study the Hamiltonian as an analytic function of Marchenko-Ostrovki parameters. Here the analysis of the Löwner equation for the quasimonentum of the corresponding Hill operator is important. We apply these results to study the action variables for the KdV equation. The similar results are obtained for non-linear Schrodinger equation.

In a joint research with Sergei Kuksin we plan to use these results to study a Birkhoff coordinate system on the space of periodic functions.

• Symplectic and Poisson structures in Hilbert spaces. Hamiltonian and Poisson. PDE. Examples.
• KAM-theory for Hamiltonian perturbations of linear equations with parameters: normal forms, iterative transformations, homological equations. Easy and difficult homological equations. Applications to equations of mathematical physics.
• KAM-theory for Hamiltonian perturbations of integrable PDE. Normal forms and reducing the problem to item 2. Discussion of the results obtained.
• KAM-theory for small solutions of nonlinear Hamiltonian PDE.
• Non-autonomous linear Hamiltonian PDE. The reducibility problem and its resolution by means of the KAM-theory.
• Open problems (easy and difficult).

The main aim of this course is to introduce some classical results of the exterior derivative d on manifolds, Lie derivatives and the Lemma of Poincaré. Then, we shall use them in the Hamiltonian formalism. Finally we shall describe the Leray-Schauder mapping degree.

The main aim of this course is to present a brief, but hopefully comprehensive, introduction to Mather's variational method for the study of convex Lagrangian and Hamiltonian systems.

First, we shall introduce the concepts of action-minimizing invariant probability measure and action-minimizing curve, prove their existence and discuss their dynamical properties. Then, we shall use them to define infinite families of invariant subsets of the phase space - i.e. the Mather sets, the Aubry sets and the Mane sets - and see in which sense they generalize KAM tori.

Finally, we shall discuss how the same results can be also obtained from the study of viscosity solutions and subsolutions of Hamilton- Jacobi equation (Fathi's weak KAM theory) or from a more symplectic point of view.

Applications to integrable and quasi-integrable Hamiltonian systems and to the study of invariant Lagrangian graphs will be presented.

The lectures will be about symplectic invariants and global periodic orbits of Hamiltonian systems on hypersurfaces of symplectic manifolds.

More precisely, we will construct the so called dynamical symplectic capacities and apply them to Hamiltonian systems. The construction will be based on a Mini-Max principle for the action functional of classical mechanics, the necessary Sobolev space techniques will be introduced.

The lectures require only standard techniques from differential geometry, functional analysis and Hamiltonian systems.

All the material presented is contained in the book: H. Hofer and E. Zehnder, Symplectic invariants and Hamiltonian Dynamics, Birkhäuser advanced texts, 1994.

General References

• Ambrosetti, A. and Prodi, G., A primer of Nonlinear Analysis, Cambridge Univ. Press, 1993.
• Nirenberg, L., Nonlinear Functional Analysis, Courant Institute of Mathematical Sciences, New York University (with an Appendix of E. Zehnder on Nash-Moser theory)
• Bourgain, J., Green's function estimates for lattice Schrödinger operators and applications}, Annals of Mathematics Studies, 158. Princeton University Press, Princeton, NJ, 2005.
• Craig, W., Problèmes de petits diviseurs dans les équations aux dérivées partielles, Panoramas et Synthèses, 9, Socièetè Mathèmatique de France, Paris, 2000.
• Kuksin, S., Hamiltonian perturbations of infinite-dimensional linear systems with imaginary spectrum, Funktsional. Anal. i Prilozhen. 21, no. 3, 22--37, 95, 1987.
• Kuksin, S., Analysis of Hamiltonian PDEs, Oxford Lecture Series in Mathematics and its Applications, 19. Oxford University Press, 2000.
• Kappeler, T. and Pöeschel, J., KAM and KdV, Springer, 2003.
• Moser J. and Zehnder E., Notes on Dynamical Systems, Courant Lecture Notes 2005, (and ICTP Lecture Notes, 1991).
• Hofer H. and Zehnder E., Symplectic invariants and Hamiltonian Dynamics, Birkhäuser advanced texts, 1994.

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