Research project on Hamiltonian systems of infinite dimension
KdV & KAM, Springer, 2003 (with Jürgen Pöschel)
Reprint: Higher Education Press, Beijing, to appear in 2010
Russian translation: KdV & KAM, Regular & Chaotic Dynamics, Moscow, 2008 Abstract: In this book we consider the Korteweg-de Vries equation u_t=u_xxx+6uu_x with periodic boundary conditions. Derived as a model equation for long surface waves in a narrow and shallow channel, this equation in fact models waves in homogeneous, weakly nonlinear and weakly dispersive media in general. Being an infinite dimensional Hamiltonian system, we construct global action-angle coordinates, which make evident that all solutions are periodic, quasi-periodic or almost-periodic, and which also lead to some new results along the way. Subsequently, these coordinates allow us to apply a general KAM theorem for Hamiltonian pdes, proving that large families of periodic and quasi-periodic solutions persist under sufficiently small Hamiltonian perturbations. The pertinent nondegeneracy conditions are verified by calculating the first few terms in the Birkhoff normal forms - an essentially elementary calculation.
The defocussing NLS equation and its normal form, preliminary version 2009, (with Benoît Grébert, Jürgen Pöschel) Abstract: In this book we consider the defocusing Nonlinear
Schrödinger equation (NLS),
iut=-uxx+2|u|^2u with periodic boundary
conditions. This equation is a nonlinear perturbation of the Schrödinger
equation for the wave function of a free particle in one space
dimension. Actually, the equation is a model for slowly varying wave
envelopes in dispersive media. Important applications are in the area of
nonlinear optics. In this book we present a concise treatment of the NLS
equation viewed as an integrable PDE. In particular, based on a
general construction of action and angle variables, we show that this
equation admits global Birkhoff coordinates in various weighted Sobolev spaces. As an application we derive
wellposedness results for the NLS equation in these spaces and prove various properties of its solutions.
Research project on regularized determinants and Witten-Laplacians
Witten-Deformation of the de Rham complex, book in preparation (with Dan Bughelea and Leonid Friedlander) Abstract: In this book we consider Witten's deformation of the de Rham complex of a closed Riemannian manifold (M,g), obtained by deforming the exterior differential d by means of a Morse function h : M →R. In a first part we study the generalized triangulation τ of M defined by the unstable manifolds of the gradient flow associated with the Morse function h and the Riemannian metric g. In a second part we show that the deformed de Rham complex splits into two subcomplexes, one of them, referred to as the small complex, being finite dimensional. It turns out that up to scaling, this complex is isomorphic to the Morse complex induced by the generalized triangulation τ. The Witten deformation of the de Rham complex can be used to compare analytic invariants of M, defined in terms of geometric data such as a Riemannian metric, with corresponding combinatorial invariants of M, defined in terms of a (generalized) triangulation. An example of such invariants is the analytic torsion and the Riemannian torsion, both in the classical as well as in the L2-setting.
Further information on current research projects: Please contact Thomas Kappeler