Bäcklund 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 Bäcklund 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. [6]] and S.Yu. Sakovich [A note in the Painlevé 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∞(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 Bäcklund–Darboux transformations.