q-Discretization of the two-dimensional Toda equations

Direct linearisation of the discrete-time two-dimensional Toda lattices

“…The mechanism of such a discretisation was explained by Date, Jimbo and Miwa [2] from the viewpoint of the theory of transformation groups, motivated by the idea in [21]. Moreover, the q-discretisation of the 2DTL, which is closely related to quantum groups, was also obtained by the bilinear approach in [17]. Another approach to discretise the 2DTL was introduced by Fordy and Gibbons [6,7], where the discrete equation arises as the superposition formula of two Bäcklund transforms for the continuous-time 2DTL on the nonlinear level.…”

Rational solutions for the discrete Painlevé II equation

“…Moreover, the discrete Airy functions which appeared in the special function type solutions for dP II can be expressed in terms of the Hermite-Weber functions [2,4], which are also the solutions of P IV [6]. Moreover, so-called the "molecular type solution" of dP II is expressed by the same τ function as that of the Bessel function type solution for P III [11]. At present, we do not know what these strange relations mean.…”

On integrability of a noncommutative q -difference two-dimensional Toda lattice equation

“…In what follows, we will prove how the bilinear Bäcklund transformation works for given τ n (x, y) and τ ′ n (x, y). For the sake of simplicity, we adopt the notations used in [30]. We denote τ ′ n (x, y) = |0, 1, .…”

“…Examples include q-difference Painéve equations, qdeformed KdV hierarchy and mKdV hierarchy, q-deformed KP hierarchy and constrained KP hierarchy, q-discrete two-dimensional Toda molecule (q-2DTM) equation and q-2DTL equation were studied as well as their integrability [25][26][27][28][29][30]. In [30], a q-2DTM equation and a q-2DTL equation as well as their determinant solutions were presented. The Bäcklund transformation and Lax pair were obtained for the former, but the Bäcklund transformation and Lax pair for the latter remains unknown.…”

“…The study of q-analogues of classical integrable systems in parallel with classical integrable systems has attracted much attention. Examples include q-difference Painéve equations, qdeformed KdV hierarchy and mKdV hierarchy, q-deformed KP hierarchy and constrained KP hierarchy, q-discrete two-dimensional Toda molecule (q-2DTM) equation and q-2DTL equation were studied as well as their integrability [25][26][27][28][29][30]. In [30], a q-2DTM equation and a q-2DTL equation as well as their determinant solutions were presented.…”