Non-commutative lattice-modified Gel’fand–Dikii systems

Abstract

We introduce integrable multicomponent non-commutative lattice systems, which can be considered as analogs of the modified Gel'fand-Dikii hierarchy. We present the corresponding systems of Lax pairs and we show directly multidimensional consistency of these Gel'fand-Dikii type equations. We demonstrate how the systems can be obtained as periodic reductions of the non-commutative lattice Kadomtsev-Petviashvilii hierarchy. The geometric description of the hierarchy in terms of Desargues maps helps to derive non-isospectral generalization of the non-commutative lattice modified Gel'fand-Dikii systems. We show also how arbitrary functions of single arguments appear naturally in our approach when making commutative reductions, which we illustrate on the non-isospectral nonautonomous versions of the lattice modified Korteweg-de Vries and Boussinesq systems.

Non-Commutative Rational Yang–Baxter Maps

“…Three-dimensional consistency of the non-commutative discrete KP map is a consequence [9] of the four-dimensional consistency of the so called Desargues maps [8]. To make connection with the Yang-Baxter maps consider N -cube graph, whose vertices are identified with binary sequences of length N with two vertices connected by an edge if their sequences differ at one place only.…”

“…with the central spectral parameter λ is the L-periodic reduction of the discrete non-commutative KP hierarchy (Gel'fand-Dikii system) linear problem (2.1) studied in [9]. …”

“…It is known [9] that the first part of Equations (2.2) implies existence of potentials ρ k such that u i,k = ρ…”

Discrete KP equation with self-consistent sources

“…Our result can be considered as a step towards the following research problems: (1) find the self-consistent source extensions by the squared-eigenfunction approach of other distinguished three dimensional integrable discrete equations and corresponding lattice maps (discrete Darboux equations and the quadrilateral lattice maps [4,14], discrete BKP equation [42], discrete CKP equation [25], quadratic reductions of quadrilateral lattices [8]) using known binary Darboux transformations for these systems; (2) find the self-consistent source extensions of distinguished reductions of the Hirota system like the discrete (modified) Korteweg-de Vries equation [45] or other discrete Gel'fand-Dikii type equations [46]; (3) using the known connection (on the continuous level) of the squared eigenfunction symmetry and the so called restricted flows of integrable hierarchies [2,60,51,52] find discrete analogs of the restricted flow equations; (4) use known algebro-geometric or analytic techniques of getting solutions to the Hirota system and its distinguished reductions find corresponding solutions of their extensions with sources; (5) find non-commutative analogs of integrable discrete equations from their known sourceless versions [48,9,12]. We would like to stress that the relation between discrete the KP equation with sources and the standard system of Hirota's discrete KP equations becomes elementary and visible on the level of discrete systems only.…”

Darboux and Binary Darboux Transformations for Discrete Integrable Systems. II. Discrete Potential mKdV Equation

“…In fact, the 2-periodic reduction method studied here has already been investigated in [2] where authors present a multidimensionally consistent hierarchy of discrete systems whose first member is the equation (1.1). Otherwise, this was refined and extended to the non-commutative case in [7]. In [2,3,4,7,17], as we see that the integrability is understood in the sense of the multidimensional consistency property, which gives a Lax pair directly.…”