Direct linearization approach to discrete integrable systems associated with ℤ _𝒩 graded Lax pairs

Abstract

Fordy and Xenitidis [ J. Phys. A: Math. Theor. 50 (2017) 165205. ( doi:10.1088/1751-8121/aa639a )] recently proposed a large class of discrete integrable systems which include a number of novel integrable difference equations, from the perspective of Z N graded Lax pairs, without providing solutions. In this paper, we establish the link between the Fordy–Xenitidis (FX) discrete systems in coprime case and linear integral equations in certain form, which reveals solution structure of these equations. The bilinear form of the FX integrable difference equations is also presented together with the associated general tau function. Furthermore, the solution structure explains the connections between the FX novel models and the discrete Gel’fand–Dikii hierarchy.

A discrete Darboux-Lax scheme for integrable difference equations

“…Since the generator of Yang-Baxter maps is a matrix refactorisation problem (13) it makes sense to understand how Bäcklund transformations for P∆Es are related to Yang-Baxter maps. In our future work, we plan to show that Yang-Baxter and entwining Yang-Baxter maps are superpositions of Bäcklund transformations of P∆Es.…”

“…Moreover, subtracting the two expressions for p 00 in (29) we end up with p10 − p01 − p00 (p 00 (q 10 − q01 ) + α − β) = 0, which holds on solutions of (14). Finally, according to Figure 2 and relation (13), the superposition principle for the auto-Bäcklund transformation (28) follows from B(q 00 , p00 ; ε 2 )B(q 00 , p00 ; ε 1 ) = B(q 00 , p00 ; ε 1 )B(q 00 , p00 ; ε 2 ), and can be written as…”

“…This led to the developement of methods for solving them, e.g. [2,13,15,24,26], classifying them, e.g. [3,7,28], analysing their integrability properties, e.g.…”

Linear integral equations and two-dimensional Toda systems

“…This is reflected in many aspects including searching for integrable discretisation of differential equations (see e.g. the review papers [20,27]), constructing integrability characteristics of nonlinear equations such as Lax pairs [7,22,36], commuting symmetries [9], and master symmetries [21,23], etc. Recently, one of the authors and his collaborator further developed the DL and established the connection between linear integral equations and the Lie algebras [8][9][10], which extended the range of application of the DL.…”

Integrable semi-discretisation of the Drinfel'd--Sokolov hierarchies

“…[5,44,45,62]), or more generally the discrete Gel'fand-Dikii (GD) hierarchy (see e.g. [7,15,18,44]) that contains higher-rank lattice equations, in addition to the discrete KP equation [4,27,41]. All these equations are associated with the A-type Lie algebras, i.e.…”