Soliton, breather, rogue wave and continuum limit in the discrete complex modified Korteweg-de Vries equation by Darboux-Bäcklund transformation

Abstract

In this paper, a discrete complex modified Korteweg-de Vries (DcmKdV) equation is investigated. Firstly, the modulational instability is studied beginning with the plane wave solution, from which we know that the discrete rogue wave may be expected as solution of the DcmKdV equation. Secondly, we propose the pseudopotential of the DcmKdV equation, from which a Darboux-Bäcklund transformation is established. Thirdly, 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, the relationship between parameters and solutions' structures is discussed and analyzed in detail. Finally, the integrable properties of the DcmKdV equation, including the Lax pair, Darboux-Bäcklund transformation and solutions, are shown for their continuous counterparts of the cmKdV equation. The method and technique used in this paper can also be extended to other nonlinear integrable equations. The results in this paper might be helpful for understanding some physical phenomena in electromagnetic waves and nonlinear optics.