**abstract:**
Near an arbitrary compact family of finite dimensional tori, left invariant under the KdV flow, we construct a real analytic, ’normal form transformation’ for the KdV equation having the following main properties:
(1) When restricted to the family of finite dimensional tori, the transformation coincides with the Birkhoff map.
(2) Up to a remainder term, which is smoothing to any given order, it is a pseudo-differential operator of order 0 in the normal directions with principal part given by the Fourier transform.
(3) It is canonical and the pullback of the KdV Hamiltonian is a paradifferential operator which is in normal form up to order three.

Such coordinates are a key ingredient for studying the stability of finite gap solutions of arbitrary size (periodic multisolitons) of the KdV equation under small, quasi-linear perturbations. This is a joint work with Thomas Kappeler.

Tue 21 May, 15:50 - 16:40, Aula Dini

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