A note on fractional KdV hierarchies

Abstract

We introduce a hierarchy of mutually commuting dynamical systems on a finite number of Laurent series. This hierarchy can be seen as a prolongation of the KP hierarchy, or a "reduction" in which the space coordinate is identified with an arbitrarily chosen time of a bigger dynamical system. Fractional KdV hierarchies are gotten by means of further reductions, obtained by constraining the Laurent series. The case of sl (2) 3 and its bihamiltonian structure is discussed in detail.

A Hidden Hierarchy for GD4

“…It can be shown (and this will be the subject of a forthcoming paper) that with every Birkhoff stratum one can associate systems of ordinary differential equations. For example to the big cell, which can be thought as the "trivial" Birkhoff stratum corresponds the usual central system studied in [2,3]. Unfortunately we cannot pursue here this general approach, so let us simply define a new particular Central System, called in the following the Hidden Central System (HCS), which actually corresponds to a Birkhoff stratum of codimension one in the Grassmanian [4].…”

“….} and for any element n in N let us define a Laurent series in L, called current, of the type 3) where N c denotes the complementary set of N in Z i.e., the set {1, −1, −2, . .…”

“…On the other hand in [2,3] it has been shown how to recover from a system of ordinary partial differential equations (called Central System (CS)) defined on the Sato Grassmanian, the GD hierarchies (as well the fractional ones) performing suitable reduction processes. It is therefore natural to wonder if there exist "Generalized Central Systems" (in what follows called Hidden Central Systems) which give rise to these new hidden hierarchies by performing the same reduction processes.…”

“…Here we want to show that this reduction process can be applied also in higher dimensions giving rise to possibly new hidden hierarchies. More precisely let us show how to construct an hidden hierarchy for the GD 4 , namely the one corresponding to the restriction to the submanifold S 4 and to the spatialization of the time t 3 . It should be noticed that in principle the first hidden hierarchy to be considered for the GD 4 hierarchy is that corresponding to the spatialization of the time t 2 and not the one which we are going to consider.…”

The soliton equations associated with the affine Kac–Moody Lie algebra G2(1)

“…Unfortunately exactly as happens for the Drinfeld-Sokolov reduction for the Lie algebra G (1) 2 the computations involved to derive the explicit expression of the bihamiltonian fields of the hierarchy are very complicated. The aim of this last section is to show how the so called Frobenius technique ( [7]) provides somehow a shortcut of the Drinfeld-Sokolov procedure.…”