Backlund transformations for new integrable hierarchies related to the polynomial Lie algebra gl∞(n)

In a recent paper [P. Casati, G. Ortenzi, New integrable hierarchies from vertex operator representations of polynomial Lie algebras, J. Geom. Phys. 56 (3) (2006) 418-449] Casati and Ortenzi gave a representation-theoretic interpretation of recently discovered coupled soliton equations, which were described by e.g. R. Hirota, X. Hu, X. Tang [A vector potential KdV equation and vector Ito equation: Soliton solutions, bilinear Backlund transformations and Lax pairs, J. Math. Anal. Appl. 288 (1) (2003) 326-348. [3]], S. Kakei [Dressing method and the coupled KP hierarchy, Phys. Lett. A 264 (6) (2000) 449-458. [611 and S.Yu. Sakovich [A note in the Painleve property of coupled KdV equation, arXiv:nlin.SI/0402004. [7]]. Casati and Ortenzi use vertex operators for these Lie algebras and a boson-fermion type of correspondence to get a hierarchy of coupled Hirota bilinear equations. In this paper we reformulate the Hirota bilinear'description for the Lie algebra gl(infinity)((n)) and obtain a bilinear identity for matrix wave functions. From that it is straightforward to deduce the Sato-Wilson, Lax and Zakharov-Shabat equations. Using these wave functions and standard calculus with vertex operators we obtain elementary Backlund-Darboux transformations. (c) 2006 Elsevier B.V. All rights reserved.