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

2013/02/28 by Adam Doliwa
Chemistry · Mathematics · Physics and Astronomy · #Algebraic structures and combinatorial models #Commutative property #Generalization #Hierarchy #Integrable system #Isospectral #Lattice (music) #Mathematical analysis #Mathematics #Molecular spectroscopy and chirality #Nonlinear Waves and Solitons #Physics #Pure mathematics #math-ph #math.DS #math.MP #nlin.SI

paper · pdf · doi:10.1088/1751-8113/46/20/205202

published as J. Phys. A: Math. Theor. 46 (2013) 205202 (14pp) · 12 pages, 1 figure; types corrected, conclusion section and new references added (v2)

arxiv created 2013/04/05 · openalex publication_date 2013/05/02 · arxiv updated 2013/08/14 · openalex created_date 2025/10/10 · openalex updated_date 2026/08/05

Abstract

We introduce integrable multicomponent non-commutative lattice systems, which can be considered as analogues of the modified Gel'fand-Dikii hierarchy.We present the corresponding systems of Lax pairs and show directly the 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 a nonisospectral 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 latticemodified Korteweg-de Vries and Boussinesq systems.

Citations

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