Algebro-Geometric Constraints on Solitons with Respect to Quasi-Periodic Backgrounds

Abstract

We investigate the algebraic conditions the scattering data of short-range perturbations of quasi-periodic finite-gap Jacobi operators have to satisfy. As our main result we provide the Poisson-Jensen-type formula for the transmission coefficient in terms of Abelian integrals on the underlying hyperelliptic Riemann surface and give an explicit condition for its single-valuedness. In addition, we establish trace formulas which relate the scattering data to the conserved quantities in this case.

arXiv comment9 pages. Bull. London Math. Soc. (to appear)

References in corpus (5)

• Sum Rules and the Szego Condition for Orthogonal Polynomials on the Real Line
• On the inverse scattering problem for Jacobi matrices with the spectrum on an interval, a finite system of intervals or a Cantor set of positive length
• Scattering Theory for Jacobi Operators with Quasi-Periodic Background
• Inverse Scattering Transform for the Toda Hierarchy with Quasi-Periodic Background
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Cited by in corpus (10)

• Stability of the periodic Toda lattice under short range perturbations
• Soliton Solutions of the Toda Hierarchy on Quasi-Periodic Backgrounds Revisited
• Stability of the Periodic Toda Lattice in the Soliton Region
• Inverse scattering transform for the Toda hierarchy with steplike finite-gap backgrounds
• Long-Time Asymptotics of Perturbed Finite-Gap Korteweg-de Vries Solutions
• Trace Formulas for Schroedinger Operators in Connection with Scattering Theory for Finite-Gap Backgrounds
• Long-Time Asymptotics for the Toda Shock Problem: Non-Overlapping Spectra
• Stability of the Periodic Toda Lattice: Higher Order Asymptotics
• A Paley-Wiener Theorem for Periodic Scattering with Applications to the Korteweg-de Vries Equation
• Trace formulas for a class of Jacobi operators