Novel periodic solutions and rogue waves of the defocusing scalar and coupled Ablowitz-Ladik systems on a nonzero background

Abstract

In this paper we apply Hirota's bilinear method to the scalar and coupled Ablowitz-Ladik systems in the defocusing dispersion regime under the assumption of a background amplitude $0<\rho<1$. We first establish, in the scalar case, the correspondence between the Hirota's parameters and the spectral parameters of the inverse scattering transform. Then we show that when the Hirota parameter associated to the discrete eigenvalue is chosen outside the range corresponding to a discrete dark soliton, novel solutions of the Ablowitz-Ladik system emerge. In general, these solutions are singular, but there exists a class of time-periodic solutions for which it is possible to choose the soliton parameters so that the breathers remain regular on the lattice for all times. We also discuss the interactions between a dark soliton and a regular breather, and between two regular breathers. For the coupled Ablowitz-Ladik system, by including in the background discrete, counter-propagating plane waves, we use Hirota's method to derive novel Akhmediev-type (i.e., space-periodic) discrete breathers which are regular for all times. Finally, taking the limit of the discrete Akhmediev breathers as the period approaches infinity (i.e., as the wavenumber approaches zero) we obtain novel rogue wave solutions of the coupled Ablowitz-Ladik system.