On τ\tau-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 $\tau$-function. We investigate the potential using the non-local $\bar\partial$ 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