J. Chen and D.E. Pelinovsky
Rogue waves arising on the standing periodic waves in the Ablowitz-Ladik equation
Stud Appl Math. 152 (2024) 147-173
Abstract:
We study the standing periodic waves in the semi-discrete integrable system modelled
by the Ablowitz-Ladik equation. We have related the stability spectrum to the Lax spectrum
by separating the variables and by finding the characteristic polynomial for the standing
periodic waves. We have also obtained rogue waves on the background of the modulationally
unstable standing periodic waves by using the end points of spectral bands and the corresponding
eigenfunctions. The magnification factors for the rogue waves have been computed
analytically and compared with their continuous counterparts. The main novelty of this work
is that we explore a non-standard linear Lax system, which is different from the standard Lax
representation of the Ablowitz-Ladik equation.
Keywords:
Ablowitz-Ladik equation, standing periodic waves,
modulational instability, rogue waves.