Estimates for periodic Zakharov–Shabat operators

Abstract

We consider the periodic Zakharov-Shabat operators on the real line. The spectrum of this operator consists of intervals separated by gaps with the lengths |g n | 0, n ∈ Z. Let μ ± n be the corresponding effective masses and let h n be heights of the corresponding slits in the quasi-momentum domain. We obtain a priori es-terms of weighted p -norms at p 1. The proof is based on the analysis of the quasi-momentum as the conformal mapping.

New Estimates of the Nonlinear Fourier Transform for the Defocusing NLS Equation

“…and is well known -see [9,22]. The case m = 1 was proved by Korotyaev [18] using conformal mapping theory, see also [20]. However, his method does not seem applicable for the case m 2.…”

“…However, his method does not seem applicable for the case m 2. In fact, it is stated as an open problem in [20]. For the case of the KdV-equation…”

“…Note that there exists a vast amount of literature on the spectral theory of these operators -see e.g. [20,10,12,6,9] and the references therein.…”

“…Even though there are only finitely many of them, they cannot be controlled by the L 2 -norm ϕ 0 as one can translate the spectrum of ϕ without changing ϕ 0 . Instead, we use the H 1 -norm ϕ 1 , and provide estimates of ϕ 1 in terms of I n by separate arguments which were inspired by work of Korotyaev [18,20].…”

Classical and Stochastic Löwner–Kufarev Equations

“…Remark 7. We also mention here relations between Löwner half-plane multi-slit equations and the estimates of spectral gaps of changing length for the periodic Zakharov-Shabat operators and for Hamiltonians in KdV and non-linear Schrödinger equations elaborated in [138,139,140]. The (chordal) stochastic Löwner evolution with parameter k ≥ 0 (SLE k ) starting at a point x ∈ R is the random family of maps (g t ) obtained from the chordal Löwner equation (5) by letting ξ (t) = √ kB t , where B t is a standard one dimensional Brownian motion such that √ kB 0 = x. Namely, let us consider the equation…”