New ideas on complete integrability

Contact complete integrability

Complete integrability in a symplectic setting means the existence of a Lagrangian foliation leaf-wise preserved by the dynamics. In the paper we describe complete integrability in a contact set-up as a more subtle structure: a flag of two foliations, Legendrian and coLegendrian, and a holonomy-invariant transverse measure of the former in the latter. This turns out to be equivalent to the exis...

New Ideas on Rehabilitation

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On the complete integrability of the discrete Nahm equations

The discrete Nahm equations, a system of matrix valued difference equations, arose in the work of Braam and Austin on half-integral mass hyperbolic monopoles. We show that the discrete Nahm equations are completely integrable in a natural sense: to any solution we can associate a spectral curve and a holomorphic linebundle over the spectral curve, such that the discrete-time DN evolution corres...

On the Complete Integrability of Some Lax Systems

On the Complete Integrability of Completely Integrable Systems

The question of complete integrability of evolution equations associated to n× n first order isospectral operators is investigated using the inverse scattering method. It is shown that for n > 2, e.g. for the three-wave interaction, additional (nonlinear) pointwise flows are necessary for the assertion of complete integrability. Their existence is demonstrated by constructing action-angle varia...