94 research outputs found
On -function of conjugate nets
We study a potential introduced by Darboux to describe conjugate nets, which
within the modern theory of integrable systems can be interpreted as a
-function. We investigate the potential using the non-local
dressing method of Manakov and Zakharov, and we show that it can
be interpreted as the Fredholm determinant of an integral equation which
naturally appears within that approach. Finally, we give some arguments
extending that interpretation to multicomponent Kadomtsev-Petviashvili
hierarchy.Comment: 8 page
Non-commutative rational Yang-Baxter maps
Starting from multidimensional consistency of non-commutative lattice
modified Gel'fand-Dikii systems we present the corresponding solutions of the
functional (set-theoretic) Yang-Baxter equation, which are non-commutative
versions of the maps arising from geometric crystals. Our approach works under
additional condition of centrality of certain products of non-commuting
variables. Then we apply such a restriction on the level of the Gel'fand-Dikii
systems what allows to obtain non-autonomous (but with central non-autonomous
factors) versions of the equations. In particular we recover known
non-commutative version of Hirota's lattice sine-Gordon equation, and we
present an integrable non-commutative and non-autonomous lattice modified
Boussinesq equation.Comment: 7 pages, 2 figures; Remark on p. 6 corrected (v2
Non-commutative q-Painleve VI equation
By applying suitable centrality condition to non-commutative non-isospectral
lattice modified Gel'fand-Dikii type systems we obtain the corresponding
non-autonomous equations. Then we derive non-commutative q-discrete Painleve VI
equation with full range of parameters as the (2,2) similarity reduction of the
non-commutative, non-isospectral and non-autonomous lattice modified
Korteweg-de Vries equation. We also comment on the fact that in making the
analogous reduction starting from Schwarzian Korteweg-de Vries equation no such
"non-isospectral generalization" is needed.Comment: 7 pages, 1 figure; introduction expanded (version 2
Non-commutative lattice modified Gel'fand-Dikii systems
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.Comment: 12 pages, 1 figure; types corrected, conclusion section and new
references added (v2
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