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

Abstract

We propose a novel semi-discrete Kadomtsev–Petviashvili equation with two discrete and one continuous independent variables, which is integrable in the sense of having the standard and adjoint Lax pairs, from the direct linearisation framework. By performing reductions on the semi-discrete Kadomtsev–Petviashvili equation, new semi-discrete versions of the Drinfel’d–Sokolov hierarchies associated with Kac–Moody Lie algebras A r ( 1 ) , A 2 r ( 2 ) , C r ( 1 ) and D r + 1 ( 2 ) are successfully constructed. A Lax pair involving the fraction of Z N graded matrices is also found for each of the semi-discrete Drinfel’d–Sokolov equations. Furthermore, the direct linearisation construction guarantees the existence of exact solutions of all the semi-discrete equations discussed in the paper, providing another insight into their integrability in addition to the analysis of Lax pairs.

Integral preserving discretization of 2D Toda lattices

“…Due to the fact that generalized Toda systems are related to many relevant mathematical theories, the number of papers on the subject is enormous: continuous Toda systems and their discretizations are addressed from different perspectives and studied using various methods. Not even attempting to make a review, we will mention the papers [7][8][9][10][11][12][13][14][15][16][17] that are important for our approach to the problem of discretizeing Toda lattices. Purely discrete versions of the generalized Toda lattice, associated with the A-series Cartan matrices, were studied in [7][8][9].…”

“…Another version of semi-discrete Toda system of the series B is obtained in [16] by considering a modification of skew-orthogonal polynomials that arise in the random matrix theory. In [17] direct linearization method was used to study semi-discrete analogs of Toda systems corresponding to some series of affine Cartan matrices. In the purely discrete case there also exist different versions of Toda systems corresponding to Cartan matrices, see [12,14].…”

“…Proof of theorem 2. Immediately follows from (17) since every y-integral annihilates the whole characteristic algebra and, in particular, it annihilates the generators X1 , . .…”