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 arbitrarily large nonzero boundary conditions at infinity. The IST was developed for this system in the past under the assumption that the amplitude of the background intensity Q0 satisfies a "small norm'' condition 0<Q0<1 [1,2] . As recently shown by
Ohta and Yang [3], the defocusing AL system, which is modulationally stable for 0<Q0<1, becomes unstable if Q0>1. And, in analogy with the focusing case [4,5], when Q0>1 the defocusing AL equation admits discrete rogue wave solutions, some of which are regular for all times [3]. We have developed the IST for the defocusing AL with Q0>1, analyzed the spectrum and characterized 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.