Self-dual Yang-Mills Hierarchy Research Articles

THE τ FUNCTIONS OF THE AKS HIERARCHY AND TWISTOR CORRESPONDENCE

Adler–Kostant–Symes scheme provides a geometrical method for constructing different integrable systems. We construct an AKS hierarchy and obtain the τ function solutions of this hierarchy. We also show that this AKS hierarchy is a reduction of self dual Yang–Mills (SDYM) equation hierarchy and discuss its twistor construction. Hence we re-establish once again that SDYM hierarchy is a universal integrable hierarchy, so that by appropriate reduction and suitable choice of gauge group this hierarchy produces all the well-known hierarchies of the soliton equations 1.

On commuting flows of AKS hierarchy and twistor correspondence

The Adler-Kostant-Symes (AKS) scheme gives a geometrical method of construction of different integrable systems. In this paper we construct an AKS hierarchy, and we show that these commuting flows are reductions of self-dual Yang-Mills hierarchy.

The twistor theory of equations of KdV type: I

This article is the first of two concerned with the development of the theory of equations of KdV type from the point of view of twistor theory and the self-dual Yang-Mills equations. A hierarchy on the self-dual Yang-Mills equations is introduced and it is shown that a certain reduction of this hierarchy is equivalent to then-generalized KdV-hierarchy. It also emerges that each flow of then-KdV hierarchy is a reduction of the self-dual Yang-Mills equations with gauge group SL n . It is further shown that solutions of the self-dual Yang-Mills hierarchy and their reductions arise via a generalized Ward transform from holomorphic vector bundles over a twistor space. Explicit examples of such bundles are given and the Ward transform is implemented to yield a large class of explicit solutions of then-KdV equations. It is also shown that the construction of Segal and Wilson of solutions of then-KdV equations from loop groups is contained in our approach as an ansatz for the construction of a class of holomorphic bundles on twistor space. A summary of the results of the second part of this work appears in the Introduction.

A self-dual Yang-Mills hierarchy and its reductions to integrable systems in 1+1 and 2+1 dimensions

The self-dual Yang-Mills equations play a central role in the study of integrable systems. In this paper we develop a formalism for deriving a four dimensional integrable hierarchy of commuting nonlinear flows containing the self-dual Yang-Mills flow as the first member. We show that upon appropriate reduction and suitable choice of gauge group it produces virtually all well known hierarchies of soliton equations in 1+1 and 2+1 dimensions and can be considered as a “universal” integrable hierarchy. Prototypical examples of reductions to classical soliton equations are presented and related issues such as recursion operators, symmetries, and conservation laws are discussed.

Symmetries of the 4D self-dual Yang-Mills equation and reduction to the 2D KdV equation

The Lie-point and the Lie-Bäcklund symmetries of 4D self-dual Yang-Mills equation are investigated as those of differential equations. The Lie-point symmetry is nothing but the gauge symmetry at the level of field equation, but the Lie-Bäcklund symmetries are new. In particular, by the symmetry reduction to KdV equation in 2D the corresponding Lie-Bäcklund symmetries which reduce to the isospectral symmetries or to the nonisospectral symmetries are identified. Some speculations on the existence of the self-dual Yang-Mills hierarchy as well as the derivation of the 4D analogue of the string equation of the nonperturbative 2D quantum gravity are given.

Symmetries of the 4D self-dual Yang-Mills equation and reduction to the 2D KdV equation

The Lie-point and the Lie-Baecklund symmetries of 4D self-dual Yang-Mills equation are investigated as those of differential equations. The Lie-point symmetry is nothing but the gauge symmetry at the level of field equation, but the Lie-Baecklund symmetries are new. In particular, by the symmetry reduction to KdV equation in 2D the corresponding Lie-Baecklund symmetries which reduce to the isospectral symmetries or to the nonisospectral symmetries are identified. Some speculations on the existence of the self-dual Yang-Mills hierarchy as well as the derivation of the 4D analogue of the string equation of the nonperturbative 2D quantum gravity are given.

Self-dual Yang–Mills hierarchy: A nonlinear integrable dynamical system in higher dimensions

A self-dual Yang–Mills (SDYM) hierarchy is presented that has some properties similar to those of the celebrated Kadomtsev–Petviashvili (KP) hierarchy having one spacial variable. The SDYM hierarchy is an infinite system of GL(∞)-invariant compatible SDYM equations depending on an infinite number of spacial variables, on which an exponential operator acts as a time evolution operator. A nontrivial GL(∞) symmetry of the SDYM hierarchy called the Bruhat transformation is found. The existence of the GL(∞) SDYM hierarchy supports the complete integrability of the single GL(∞)SDYM equation with a reduction condition, however, our observation suggests that the SDYM equation without the condition may not be completely integrable in sharp contrast with the KP equation.

Hierarchy structure in integrable systems of gauge fields and underlying Lie algebras

An improved version of Nakamura's self-dual Yang-Mills hierarchy is presentd and its symmetry contents are studied. The new hierarchy as well as the previous one represents a set of commuting dynamical flows in an infinite dimensional manifolds of “loop type”, but includes a large set of dependent variables. Because of new degrees of freedom the theory acquires a more symmetric form with richer structures. For example it allows a large symmetry algebra of Riemann-Hilbert type, which is actually a direct sum of two subalgebras (“left” and “right”). This phenomenon is basically the same as observed recently by Avan and Bellon on the case of principal chiral models. In addition to these rather familiar symmeties, a new type of symmetries referred to as “coordinate transformation type” are also introduced. Generators of the above dynamical flows are all included therein. These two types of symmetries altogether form a big Lie algebra, which lead to more satisfactory understanding of symmetry properties of integrable systems of guage fields.

On self-dual yang-Mills hierarchy

In this note, motivated by the Kadomtsev-Petviashvili (KP) hierarchy of integrable nonlinear evolution equations, a GL(n, C) self-dual Yang-Mills (SDYM) hierarchy is presented; it is an infinite system of SDYM equations having an infinite number of independent variables and being outside of the KP hierarchy. A relationship between the KP hierarchy and the SDYM hierarchy is discussed. It is also shown that GL(∞) SDYM equations introduced in this note are reduced to the GL(n, C) SDYM hierarchy by imposing an algebraic constraiint.

Periodic reduction of self-dual Yang–Mills equations

A class of GL(∞)-invariant self-dual Yang–Mills (SDYM) equations is considered. It is shown that the GL(∞) SDYM equations are reduced to the GL(n,C) SDYM hierarchy by imposing a periodic condition. This reduction procedure makes clear a relationship between our GL(∞) SDYM equations and the usual infinite matrix representation of a single GL(n,C) SDYM equation.

Transformation group acting on a self-dual Yang–Mills hierarchy

A GL(n,C) self-dual Yang–Mills hierarchy is introduced; it is an infinite system of self-dual Yang–Mills equations having an infinite number of independent variables. Cauchy problems for the hierarchy are formally solved by using Lie transforms of a wave matrix. A relationship between the Kadomtsev–Petviashvili hierarchy and the self-dual Yang–Mills hierarchy is discussed. Furthermore, it is shown that an infinite-dimensional transformation group acts on a solution space to the (n≥2) self-dual Yang–Mills hierarchy. A parametric solution to the hierarchy is also given as a representation of the transformation group.