A discrete Darboux–Lax scheme for integrable difference equations

• Published in: Chaos, solitons and fractals Vol. 158; p. 112059
• Main Authors: Fisenko, X., Konstantinou-Rizos, S., Xenitidis, P.
• Format: Journal Article
• Language: English
• Published: Elsevier Ltd 01.05.2022
• Subjects: 3D-consistency; Bäcklund transformations; Darboux transformations; Integrable lattice equations; Partial difference equations; Quad-graph equations; Soliton solutions; Darboux transformations; 3D-consistency; Bäcklund transformations; Partial difference equations; Integrable lattice equations; Soliton solutions; Quad-graph equations
• ISSN: 0960-0779; 1873-2887
• DOI: 10.1016/j.chaos.2022.112059

We propose a discrete Darboux–Lax scheme for deriving auto-Bäcklund transformations and constructing solutions to quad-graph equations that do not necessarily possess the 3D consistency property. As an illustrative example we use the Adler–Yamilov type system which is related to the nonlinear Schrödinger (NLS) equation [7]. In particular, we construct an auto-Bäcklund transformation for this discrete system, its superposition principle, and we employ them in the construction of the one- and two-soliton solutions of the Adler–Yamilov system. 02.30.Ik, 02.90.+p, 03.65.Fd 37K60, 39A36, 35Q55, 16T25. •A new method for solving integrable nonlinear partial diference equations (PΔEs)•Systematic derivation of soliton solutions to integrable nonlinear PΔEs•Study of an Adler-Yamilov type of system which discretises the NLS equation•Construction of Darboux and Bäcklund transformations for the Adler-Yamilov system•Derivation of one- and two-soliton solutions of the Adler-Yamilov system