Soliton, breather, rogue wave and continuum limit for the spatial discrete Hirota equation by Darboux-Backlund transformation

In this paper, the spatial discrete Hirota equation is investigated by Darboux-Backlund transformation. Firstly, the pseudopotential of the spatial discrete Hirota equation is proposed for the first time, from which a Darboux-Backlund transformation is constructed. Comparing it with the corresponding onefold Darboux transformation, we find that they are equivalent because there is no difference except for a constant times. We believe that this equivalence may hold universal if these two transformations are all derived from the same discrete spectral problem and using the similar technique in the references. Secondly, starting from vanishing and plane wave backgrounds, a variety of nonlinear wave solutions, including bell-shaped one-soliton, three types of breathers, W-shaped soliton, periodic solution and rogue wave are given, and the relevant dynamical properties and evolutions are illustrated by plotting figures. The relationship between parameters and solutions' structures is studied in detail, and the related method and technique can also be extended to other nonlinear integrable equations. Finally, we show that the continuum limit of breather and rogue wave solutions of the spatial discrete Hirota equation yields the counterparts of the Hirota equation. The results in this paper might be useful for understanding some physical phenomena in nonlinear optics.