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Perturbations of the harmonic map equation

Thomas Kappeler — 2002

Journées équations aux dérivées partielles

We consider perturbations of the harmonic map equation in the case where the source and target manifolds are closed riemannian manifolds and the latter is in addition of nonpositive sectional curvature. For any semilinear and, under some extra conditions, quasilinear perturbation, the space of classical solutions within a homotopy class is proved to be compact. For generic perturbations the set of solutions is finite and we present a count of this set. An important ingredient for our analysis is...

Fibration of the phase space for the Korteweg-de Vries equation

Thomas Kappeler — 1991

Annales de l'institut Fourier

In this article we prove that the fibration of L 2 ( S 1 ) by potentials which are isospectral for the 1-dimensional periodic Schrödinger equation, is trivial. This result can be applied, in particular, to N -gap solutions of the Korteweg-de Vries equation (KdV) on the circle: one shows that KdV, a completely integrable Hamiltonian system, has global action-angle variables.

Symmetries of the nonlinear Schrödinger equation

Benoît GrébertThomas Kappeler — 2002

Bulletin de la Société Mathématique de France

Symmetries of the defocusing nonlinear Schrödinger equation are expressed in action-angle coordinates and characterized in terms of the periodic and Dirichlet spectrum of the associated Zakharov-Shabat system. Application: proof of the conjecture that the periodic spectrum < λ k - λ k + < λ k + 1 - of a Zakharov-Shabat operator is symmetric,. λ k ± = - λ - k for all k , if and only if the sequence ( γ k ) k of gap lengths, γ k : = λ k + - λ k - , is symmetric with respect to k = 0 .

Comparison of the refined analytic and the Burghelea-Haller torsions

Maxim BravermanThomas Kappeler — 2007

Annales de l’institut Fourier

The refined analytic torsion associated to a flat vector bundle over a closed odd-dimensional manifold canonically defines a quadratic form τ on the determinant line of the cohomology. Both τ and the Burghelea-Haller torsion are refinements of the Ray-Singer torsion. We show that whenever the Burghelea-Haller torsion is defined it is equal to ± τ . As an application we obtain new results about the Burghelea-Haller torsion. In particular, we prove a weak version of the Burghelea-Haller conjecture relating...

Inverse spectral results on even dimensional tori

Carolyn S. GordonPierre GueriniThomas KappelerDavid L. Webb — 2008

Annales de l’institut Fourier

Given a Hermitian line bundle L over a flat torus M , a connection on L , and a function Q on M , one associates a Schrödinger operator acting on sections of L ; its spectrum is denoted S p e c ( Q ; L , ) . Motivated by work of V. Guillemin in dimension two, we consider line bundles over tori of arbitrary even dimension with “translation invariant” connections , and we address the extent to which the spectrum S p e c ( Q ; L , ) determines the potential Q . With a genericity condition, we show that if the connection is invariant under...

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