Linear integral equations and two‐dimensional Toda systems

Abstract

The direct linearization framework is presented for the two‐dimensional (2D) Toda equations associated with the infinite‐dimensional Lie algebras A∞, B∞, and C∞, as well as the Kac–Moody algebras Arfalse(1false), A2rfalse(2false), Crfalse(1false), and Dr+1false(2false) for arbitrary integers r∈double-struckZ+, from the aspect of a set of linear integral equations in a certain form. Such a scheme not only provides a unified perspective to understand the underlying integrability structure, but also induces the direct linearizing type solution potentially leading to the universal solution space, for each class of the 2D Toda system. As particular applications of this framework to the 2D Toda lattices, we rediscover the Lax pairs and the adjoint Lax pairs and simultaneously construct the generalized Cauchy matrix solutions.

Integrable semi-discretisation of the Drinfel’d–Sokolov hierarchies

“…finite-pole solution) can be easily obtained by taking a special integration measure containing a finite number of poles. Since the reductions that we perform in this paper coincide with those in the theory of the 2D Toda-type equations, we refer the reader to [61] for the general formulae of the Cauchy matrix solutions of all the discussed semi-discrete Drinfel'd-Sokolov equations by substituting the plane wave factors ρ n (k) and σ n (k ) with (1.3).…”

“…In this section, we construct the semi-discrete equations associated with infinite-dimensional algebras B ∞ and C ∞ and also Kac-Moody algebras A (1) r , A (2) 2r , C (1) r and D (2) r+1 in the Drinfel'd-Sokolov classification, from the semi-discrete equation (3.6). This is realised by imposing restrictions on the integration measure dζ(k, k ) and the integration domain D (see [61]), which leads to symmetry and periodicity constraints on the discrete independent variable n. As a consequence, the independent variable n turns out to be an index, labelling multi-component variables for the reduced integrable equations.…”

“…In the recent papers [21,61], the connection between the linear integral equations in the DL framework and the continuous Drinfel'd-Sokolov and 2D Toda hierarchies associated with the infinite-dimensional algebras A ∞ , B ∞ and C ∞ as well as the Kac-Moody Lie algebras A (1) r , A (2) 2r , C (1) r and D (2) r+1 was established. This gives us a strong hint about how to construct the discrete Drinfel'd-Sokolov hierarchies of these types from the DL framework.…”

On the Fourth-Order Lattice Gel'fand-Dikii Equations

“…Soon after the discrete DL scheme was developed [30,33]. And then, this method has been widely used to investigate continuous, semidiscrete, and discrete integrable systems [11,12,13,14,28,30,33,34,41,44]. In this paper, we are interested in the construction of the fourth-order lattice GD type equations as well as their MDC property and Lax integrability.…”

On the fourth-order lattice Gel'fand-Dikii equations

“…Soon after the discrete DL scheme was developed [12,22]. And then, this method has been widely used to investigate continuous, semidiscrete, and discrete integrable systems [12,20,[22][23][24][25][26][27][28][29]. In this paper, we are interested in the construction of the fourth-order lattice GD type equations as well as their MDC property and Lax integrability.…”