Multicomponent KP type hierarchies and their reductions, associated to conjugacy classes of Weyl groups of classical Lie algebras

Abstract

This, to a large extent, expository paper describes the theory of multicomponent hierarchies of evolution equations of XKP type, where X = A, B, C, or D, and AKP = KP and their reductions, associated with the conjugacy classes of the Weyl groups of classical Lie algebras of type X. As usual, the main tool is the multicomponent boson–fermion correspondence, which leads to the corresponding tau-functions, wave functions, dressing operators, and Lax operators.

Lax representation and Bäcklund–Darboux transformation for coupled BKP hierarchy

“…Coupled BKP hierarchy is a special case of matrix KP hierarchy 14 with Lax operator's coefficients being matrices, which is related with multi-component KP theory. 4,5,10 Therefore, we believe it will be very interesting to discuss the applications of polynomial Lie algebras in multi-component KP theory.…”

“…Later Date et al 3 gave a construction of the Kadomtsev-Petviashvili (KP) and Korteweg-de Vries (KdV) hierarchies in terms of the vertex operator representation of affine Lie algebras. Further, Kac and his students gave more important works on this field, [4][5][6] and in particular Kac-Wakimoto construction 6 was proposed to construct integrable hierarchies of any affine Lie algebras. Here in this paper, we will concentrate on the integrable hierarchies related to polynomial Lie algebras.…”