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 non-autonomous versions of the lattice modified Korteweg-de Vries and Boussinesq systems.

12 pages, 1 figure; types corrected, conclusion section and new references added (v2)

References in corpus (9)

• Boussinesq-like multi-component lattice equations and multi-dimensional consistency
• Quantum geometry of 3-dimensional lattices
• The affine Weyl group symmetry of Desargues maps and of the non-commutative Hirota-Miwa system
• An integrable multicomponent quad equation and its Lagrangian formulation
• Nonlinear PDEs for Fredholm determinants arising from string equations
• Quantum Painlevé systems of type $A_l^{(1)}$
• Quantization of three-wave equations
• Hirota equation and the quantum plane
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Cited by in corpus (16)

• Non-commutative rational Yang-Baxter maps
• On non-abelian quadrirational Yang-Baxter maps
• Discrete KP equation with self-consistent sources
• Non-commutative bi-rational maps satisfying Zamolodchikov equation, and Desargues lattices
• The Coxeter relations and KP map for non-commuting symbols
• Darboux and Binary Darboux Transformations for Discrete Integrable Systems. II. Discrete Potential mKdV Equation
• Non-commutative Hermite--Padé approximation and integrability
• Non-Abelian hierarchies of compatible maps, associated integrable difference systems and Yang-Baxter maps
• Symmetries and Special Solutions of Reductions of the Lattice Potential KdV Equation
• Non-commutative double-sided continued fractions
• Direct linearisation approach to discrete integrable systems associated with ${\mathbb{Z}}_{\mathcal{N}}$ graded Lax pairs
• Desargues maps and their reductions
• Integrable semi-discretisation of the Drinfel'd--Sokolov hierarchies
• Discrete Lax pairs and hierarchies of integrable difference systems
• Quadrangular sets in projective line and in Moebius space, and geometric interpretation of the non-commutative discrete Schwarzian Kadomtsev-Petviashvili equation
• Non-commutative q-Painleve VI equation