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L. A. Dickey

Publications and source records attributed to L. A. Dickey.

13 recordsLinked to original sources
arXiv2004

On the Treves Criterion for the Boussinesq and other GD Hierarchies

This is an addition to our paper (Lett.Math.Phys 65,2003,187-197) where an analogue of the Treves criterion for the first integrals of the KdV hierarchy was suggested for the Boussinesq hierarchy and its necessity was proven. In the present paper, it is proven that there exists the second, ``conjugated'', group of tests. Besides, the relationship between our method and the Bäcklund transformation method recently suggested by Morosi and Pizzocchero is discussed.

nlin.SI↗
arXiv2003

On Treves' Algebraic Characterization of the KdV Hierarchy

We have found a possibility to streamline the proof of the Treves' theorem (Duke Math. J., 108, 251-295, 2001) on an algebraic characterization of the KdV hierarchy which makes it significantly shorter, following essentially the logic of the original proof.

nlin.SI↗
arXiv2000

Chains of KP, Semi-infinite 1-Toda Lattice Hierarchy and Kontsevich Integral

There are well-known constructions of integrable systems which are chains of infinitely many copies of the equations of the KP hierarchy ``glued'' together with some additional variables, e.g., the modified KP hierarchy. Another interpretation of the latter, in terms of infinite matrices, is called the 1-Toda lattice hierarchy. One way infinite reduction of this hierarchy has all solutions in the form of sequences of expanding Wronskians. We define another chain of the KP equations, also with solutions of the Wronsksian type, which is characterized by the property to stabilize with respect to a gradation. Under some constraints imposed, the tau functions of the chain are the tau functions associated with the Kontsevich integrals.

nlin.SI↗
arXiv1997

Poisson brackets with divergence terms in field theories: two examples

In field theories one often works with the functionals which are integrals of some densities. These densities are defined up to divergence terms (boundary terms). A Poisson bracket of two functionals is also a functional, i.e., an integral of a density. Suppose the divergence term in the density of the Poisson bracket be fixed so that it becomes a bilinear form of densities of two functionals. Then the left-hand side of the Jacobi identity written in terms of densities is not necessarily zero but a divergence of a trilinear form. The question is: what can be said about this trilinear form, what kind of a higher Jacobi identity (involving four fields) it enjoys? Two examples whose origin is the theory of integrable systems are given.

solv-int↗
arXiv1996

On a generalization of the Fay-Sato identity for KP Baker functions and its application to constrained hierarchies

Some new formulas for the KP hierarchy are derived from the differential Fay identity. They proved to be useful for the $k$-constrained hierarchies providing a series of determinant identities for them. A differential equation is introduced which is called ``universal" since it plays an important role for all the $k$-constrained hierarchies. In the cases $k=1,2$ and 3 explicit formulas are presented, in all the others recurrence relations are given which enable one to obtain the identities.

solv-int↗
arXiv1995

On tau-functions of Zakharov-Shabat and other matrix hierarchies of integrable equations

Matrix hierarchies are: multi-component KP, general Zakharov-Shabat (ZS) and its special cases, e.g., AKNS. The ZS comprises all integrable systems having a form of zero-curvature equations with rational dependence of matrices on a spectral parameter. The notion of a $τ$-function is introduced here in the most general case along with formulas linking $τ$-functions with wave Baker functions. The method originally invented by Sato et al. for the KP hierarchy is used. This method goes immediately from definitions and does not require any assumption about the character of a solution, being the most general. Applied to the matrix hierarchies, it involves considerable sophistication. The paper is self-contained and does not expect any special prerequisite from a reader.

hep-th↗
arXiv1994

On the constrained KP hierarchy II

A constrained KP hierarchy is discussed that was recently suggested by Aratyn et al. and by Bonora et al. This hierarchy is a restriction of the KP to a submanifold of operators which can be represented as a ratio of two purely differential operators of prescribed orders. Explicit formulas for action of vector fields on these two differential operators are written which gives a new description of the hierarchy and provides a new, more constructive, proof of compatibility of the constraint with the hierarchy. Also the Poisson structure of the constrained hierarchy is discussed.

hep-th↗
arXiv1994

On the constrained KP hierarchy

An explanation for the so-called constrained hierarhies is presented by linking them with the symmetries of the KP hierarchy. While the existence of ordinary symmetries (belonging to the hierarchy) allows one to reduce the KP hierarchy to the KdV hierarchies, the existence of additional symmetries allows to reduce KP to the constrained KP.

hep-th↗
arXiv1993

Why the general Zakharov-Shabat equations form a hierarchy?

The totality of all Zakharov-Shabat equations (ZS), i.e., zero-curvature equations with rational dependence on a spectral parameter, if properly defined, can be considered as a hierarchy. The latter means a collection of commuting vector fields in the same phase space. Further properties of the hierarchy are discussed, such as additional symmetries, an analogue to the string equation, a Grassmannian related to the ZS hierarchy, and a Grassmannian definition of soliton solutions.

hep-th↗
arXiv1992

Additional symmetries of KP, Grassmannian, and the string equation II

As in the first part of this paper (hep-th 9204092), solutions to a string equation are regarded as fixed points of some additional symmetries of a hierarchy of integrable equations. In this part matrix hierarchies are studied: the multi-component KP and KdV hierarchies, and the modified KdV hierarchy as their reduction. In particular, the action of additional symmetries on the Grassmannian is found, as well as Virasoro constraints on the $τ$-functions. The matrix string equations are known to be involved in some matrix models.

hep-th↗