Asymptotics of Spectral Data for Potentials on a Horizontal Strip

Abstract

As a final result, we study the asymptotic behavior of the spectral data (Σ, D) corresponding to a simply periodic solution $u: X \to \mathbb {C}$ of the sinh-Gordon equation defined on an entire horizontal strip $X \subset \mathbb {C}$ with positive height. Because such a solution is real analytic on the interior of X, we expect a far better asymptotic for such spectral data than for the spectral data of Cauchy data potentials (u, u _y) with only the weak requirements u ∈ W ^{1, 2}([0, 1]), u _y ∈ L ²([0, 1]) we have been using throughout most of the paper. More specifically, we expect both the distance of branch points ϰ_k,1 − ϰ_k,2 of the spectral curve Σ and the distance of the corresponding spectral divisor points to the branch points to fall off exponentially for k →±∞.