τ functions and zero curvature equations of Toda–AKNS type

Abstract

The connection between τ functions and zero curvature equations for the homogeneous construction of the basic module L(Λ0) over the simplest affine Kac–Moody algebra A(1)1 is studied.

Symmetries in the third Painlevé equation arising from the modified Pohlmeyer-Lund-Regge hierarchy

“…In this section we construct formal solutions of these equations based on the theory of the affine Lie algebraŝl 2 and its group [1], [6]. This is equivalent to a modification of the 2-component Toda lattice hierarchy [20].…”

“…In what follows we assume that c 3 c 4 = 0 (D (1) 6 -type), in which case the number of parameters contained in (1.2) is two by means of a suitable change of scales for y and s. It is known that the transformation group of solutions of (1.2) is isomorphic to the affine Weyl group of type A (1) 1 × A (1) 1 (or B (1) 2 ) [16].…”

“…In [15] Noumi and Yamada proposed a systematic description of nonlinear differential equations comprising the second, fourth and fifth Painlevé equations which possess A (1) n affine Weyl group symmetry. These systems are obtained by similarity reductions from the A (1) n Drinfeld-Sokolov hierarchy which is an infinite-dimensional integrable system characterized by affine Lie algebras and their Heisenberg subalgebras [2], [5].…”

“…These systems are obtained by similarity reductions from the A (1) n Drinfeld-Sokolov hierarchy which is an infinite-dimensional integrable system characterized by affine Lie algebras and their Heisenberg subalgebras [2], [5]. Also, Fuji and Suzuki derived dynamical systems, including the sixth Painlevé equation, by similarity reductions from the D (1) 2n+2 Drinfeld-Sokolov hierarchy [3]. Therefore, the affine Weyl group symmetries in the Painlevé II, IV, V, VI equations can all be derived from integrable systems associated with affine Lie algebra.…”

The dressing techniques for intermediate hierarchies

“…From (19) and (15) follow that Poisson bracket (11) for the functionals ℓ X and ℓ Y coincides with the second Gelfand-Dickey bracket [10] …”