Speaker: Barbara Prinari, University of Colorado, Colorado Springs
Date: April 25, 2019
Title: Inverse scattering transform for the defocusing Ablowitz-Ladik equation with arbitrary nonzero background
In this talk we discuss the inverse scattering transform (IST) for the defocusing Ablowitz-Ladik equation with an arbitrarily large nonzero background. The IST was developed in the past [1, 2] under the assumption that the amplitude of the background intensity Qo satisfies a “small norm” condition 0 < Qo < 1. As recently shown by Ohta and Yang , the defocusing AL system, which is modulationally stable for 0 ≤ Qo < 1, becomes unstable if Qo > 1. And, in analogy with the focusing case, when Qo > 1 the defocusing AL equation admits discrete rogue wave solutions, some of which are regular for all times. Therefore, it is clearly of importance to develop the IST for the defocusing AL with Qo > 1, analyze the spectrum and characterize the soliton and rational solutions from a spectral point of view. Both the direct and the inverse problems are formulated in terms of a suitable uniform variable; the inverse problem is posed as a Riemann-Hilbert problem in the complex plane, and solved by properly accounting for the asymptotic dependence of eigenfunctions and scattering data on the Ablowitz-Ladik potential.
 V.E. Vekslerchik and V.V. Konotop, Discrete nonlinear Schrodinger equation under non-vanishing boundary conditions, Inv. Probl., 8, (1992) 889.
 M.J. Ablowitz, G. Biondini and B. Prinari, Inverse scattering transform for the integrable discrete nonlinear Schrodinger equation with non- vanishing boundary conditions, Inv. Probl., 23, (2007) 1711.
 Y. Ohta and J. Yang, General rogues waves in the focusing and defocusing Ablowitz-Ladik equations, J. Phys. A, 47, (2014) 255201.