H. Aratyn

Exact static soliton solutions of (3+1)-dimensional integrable theory with nonzero Hopf numbers

In this paper we construct explicitly an infinite number of Hopfions (static, soliton solutions with non-zero Hopf topological charges) within the recently proposed 3+1-dimensional, integrable and relativistically invariant field theory. Two integers label the family of Hopfions we have found. Their product is equal to the Hopf charge which provides a lower bound to the solitons finite energy. The Hopfions are constructed explicitly in terms of the toroidal coordinates and shown to have a form of linked closed vortices.

Method of Squared Eigenfunction Potentials in Integrable Hierarchies of KP Type

Abstract:The method of squared eigenfunction potentials (SEP) is developed systematically to describe and gain new information about the Kadomtsev–Petviashvili (KP) hierarchy and its reductions. Interrelation to the τ-function method is discussed in detail. The principal result, which forms the basis of our SEP method, is the proof that any eigenfunction of the general KP hierarchy can be represented as a spectral integral over the Baker–Akhiezer (BA) wave function with a spectral density expressed in terms of SEP. In fact, the spectral representations of the (adjoint) BA functions can, in turn, be considered as defining equations for the KP hierarchy. The SEP method is subsequently used to show how the reduction of the full KP hierarchy to the constrained KP (cKPrm) hierarchies can be given entirely in terms of linear constraint equations on the pertinent τ-functions. The concept of SEP turns out to be crucial in providing a description of cKPrm hierarchies in the language of the universal Sato Grassmannian and finding the non-isospectral Virasoro symmetry generators acting on the underlying τ-functions. The SEP method is used to write down generalized binary Darboux-Bäcklund transformations for constrained KP hierarchies whose orbits are shown to correspond to a new Toda model on a square lattice. As a result, we obtain a series of new determinant solutions for the τ-functions generalizing the known Wronskian (multi-soliton) solutions. Finally, applications to random matrix models in condensed matter physics are briefly discussed.

Kac-Moody construction of Toda type field theories

Abstract Using the coadjoint orbit method we derive a geometric WZWN action based on the extended two-loop Kac-Moody algebra. We show that under a hamiltonian reduction procedure, which respects conformal invariance, we obtain a hierarchy of Toda type field theories, which contain as submodels the Toda molecule and periodic Toda lattice theories. We also discuss the classical r -matrix and integrability properties.

Toroidal solitons in 3 + 1 dimensional integrable theories

University of Illinois at Chicago, Department of Physics, 845 W. Taylor St., Chicago, IL 60607-7059

Constrained KP Hierarchies: Additional Symmetries, Darboux–Bäcklund Solutions and Relations to Multi-Matrix Models

This paper provides a systematic description of the interplay between a specific class of reductions denoted as cKPr,m(r,m ≥ 1) of the primary continuum integrable system — the Kadomtsev–Petviashvili (KP) hierarchy and discrete multi-matrix models. The relevant integrable cKPr,m structure is a generalization of the familiar r-reduction of the full KP hierarchy to the SL(r) generalized KdV hierarchy cKPr,0. The important feature of cKPr,m hierarchies is the presence of a discrete symmetry structure generated by successive Darboux–Backlund (DB) transformations. This symmetry allows for expressing the relevant tau-functions as Wronskians within a formalism which realizes the tau-functions as DB orbits of simple initial solutions. In particular, it is shown that any DB orbit of a cKPr,1 defines a generalized two-dimensional Toda lattice structure. Furthermore, we consider the class of truncated KP hierarchies (i.e. those defined via Wilson–Sato dressing operator with a finite truncated pseudo-differential series) and establish explicitly their close relationship with DB orbits of cKPr,m hierarchies. This construction is relevant for finding partition functions of the discrete multi-matrix models. The next important step involves the reformulation of the familiar nonisospectral additional symmetries of the full KP hierarchy so that their action on cKPr,m hierarchies becomes consistent with the constraints of the reduction. Moreover, we show that the correct modified additional symmetries are compatible with the discrete DB symmetry on the cKPr,m DB orbits.

Hirota's solitons in the affine and the conformal affine Toda models

Department of Physics University of Illinois at Chicago, 801 W. Taylor Street, Chicago, IL 60607-7059

On negative flows of the AKNS hierarchy and a class of deformations of a bihamiltonian structure of hydrodynamic type

A deformation parameter of a bihamiltonian structure of hydrodynamic type is shown to parametrize different extensions of the AKNS hierarchy to include negative flows. This construction establishes a purely algebraic link between, on the one hand, two realizations of the first negative flow of the AKNS model and, on the other, two-component generalizations of Camassa–Holm- and Dym-type equations. The two-component generalizations of Camassa–Holm- and Dym-type equations can be obtained from the negative-order Hamiltonians constructed from the Lenard relations recursively applied on the Casimir of the first Poisson bracket of hydrodynamic type. The positive-order Hamiltonians, which follow from the Lenard scheme applied on the Casimir of the second Poisson bracket of hydrodynamic type, are shown to coincide with the Hamiltonians of the AKNS model. The AKNS Hamiltonians give rise to charges conserved with respect to equations of motion of two-component Camassa–Holm- and two-component Dym-type equations.

On two-current realization of KP hierarchy

Abstract A simple description of the KP hierarchy and its multi-hamiltonian structure is given in terms of two Bose currents. A deformation scheme connecting various W-infinity algebras and the relation between two fundamental nonlinear structures are discussed. Properties of Faa di Bruno polynomials are extensively explored in this construction. Applications of our method are given for the Conformal Affine Toda model, WZNW models and discrete KP approach to Toda lattice chain.

R-matrix formulation of KP hierarchies and their gauge equivalence

Abstract The Adler-Kostant-Symes R -bracket scheme is applied to the algebra of pseudodifferential operators to relate the three integrable hierarchies: KP and its two modifications, known as non-standard integrable models. All three hierarchies are shown to be equivalent and a connection is established in the form of a symplectic gauge transformation. This construction results in a new representation of the W-infinity algebras in terms of four boson fields.

Virasoro symmetry of constrained KP hierarchies

Abstract The conventional formulation of additional nonisospectral sysmmetries for the full Kadomtsev-Petviashvili (KP) integrable hierarchy is not compatible with the reduction to the important class of constrained KP (cKP) integrable models. This paper solves explicitly the problem of compatibility of the Virasoro part of additional symmetries with the underlying constraints of cKP hierarchies. Our construction involves an appropriate modification of the standard additional-symmetry flows by adding a set of “ghost symmetry” flows. We also discuss the special case of cKP — truncated KP hierarchies, obtained as Darboux-Backlund orbits of initial purely differential Lax operators. Our construction establishes the condition for commutativity of the additional-symmetry flows with the discrete Darboux-Backlund transformations of cKP hierarchies leading to a new derivation of the string-equation constraint in matrix models.