On the symplectic phase space of K\lowercase{d}V

Abstract

Abstract. We prove that the Birkhoff map Ω for KdV constructed on H −1 0 (T) can be interpolated between H −1 0 (T) and L 2 0 (T). In particular, the symplectic phase space H

On the characterization of the smoothness of skew-adjoint potentials in periodic Dirac operators

“…See also the Epilog in [18]. In [14], Kappeler, Serier, and Topalov, using still another approach involving flows related to angle variables, presented a very short proof of the analogous result for potentials in certain spaces of distributions.…”

“…Concerning Dirac operators see [2,4]. Most of the results mentioned in this short survey of recent advances in this topic -but not the results obtained in [14] -are discussed in the survey article [5].…”

On the wellposedness of the KdV equation on the space of pseudomeasures

“…In a next step we study the P -equation A λ u = P n V (u+v) of (8). For n n s (q), u ∈ P n , and λ ∈ S n , substitute in it the solutionv u,λ of (11), given by (15),…”

“…[10,15]) There exists a complex neighborhood W of H −1 0 within H −1 0,C and an analytic map Φ : W → h −1/2 0,C , q → (z n (q)) n∈Z with the following properties:(i) Φ is canonical in the sense that {z n , z −n } = ∂ u z n ∂ x ∂ u z −n dx = i for all n 1,whereas all other brackets between coordinate functions vanish. (ii) For any s −1, the restriction Φ| H s 0 is a map Φ| H s 0 : H s 0 → h s+1/2 0 which is a bianalytic diffeomorphism.…”

On the Convexity of the KdV Hamiltonian

“…In [11] and [13], the restrictions of the Birkhoff map Φ : H −1 0 → −1/2,2 0 , q → (z n (q)) n∈Z , z 0 (q) = 0, to the Sobolev spaces H s 0 , −1 s 0, are studied. It turns out that the arguments developed in these two papers can also be adapted to prove Theorem 1.2.…”