On normalized differentials on hyperelliptic curves of infinite genus

Abstract

We develop a new approach for constructing normalized differentials on hyperelliptic curves of infinite genus and obtain uniform asymptotic estimates for the distribution of their zeros.

Arnold-Liouville theorem for integrable PDEs: a case study of the focusing NLS equation

“…[2,4] for the case of finite gap potentials as well as [11,28,29,8] for the case of more general potentials in H N r ). Such normalized differentials (with properties needed for our purposes) for generic potentials in iH N r have been constructed in [22] (cf. also [12]).…”

“…Note that the case of potentials in iH N r is more complicated since the operator L(ϕ) is not selfadjoint. An important ingredient for estimates, needed to construct the angle coordinates, is the rather precise localization of the zeros of these differentials provided in [22]. We emphasize that no assumptions are made on the Dirichlet eigenvalues of L(ϕ).…”

“…Let us begin with the one forms. To obtain such a family of one forms, we apply Theorem 1.3 in [22]: shrinking the ball U s if necessary, it follows that there exist analytic functions…”

On nomalized differentials on spectral curves associated to thesinh-Gordon equation

“…Such a confinement no longer holds in the case of the spectral curves associated to the focusing NLS equation, since in this case the corresponding Lax operator is not self-adjoint. To handle this case, a quite different approach was developed in [7] (cf. also [5]).…”

On nomalized differentials on spectral curves associated with the sinh-Gordon equation

“…Such a confinement no longer holds in the case of the spectral curves associated to the focusing NLS equation, since in this case the corresponding Lax operator is not self-adjoint. To handle this case, a quite different approach was developed in [17] (cf. also [13]).…”