Svetlana Pacheva

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.

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.

String and brane models with spontaneously or dynamically induced tension

We study in some detail the properties of a previously proposed new class of string and brane models whose world-sheet (world-volume) actions are built with a modified reparametrization-invariant measure of integration and which do not contain any ad hoc dimensionfull parameters. The ratio of the new and the standard Riemannian integration measure densities plays the role of a dynamically generated string/brane tension. The latter is identified as (the magnitude of) an effective (non-Abelian) electric field-strength on the world-sheet/world-volume obeying the standard Gauss-law constraint. As a result a simple classical mechanism for confinement via strings is proposed.

Emergent Cosmology, Inflation and Dark Energy

A new class of gravity–matter models defined in terms of two independent non-Riemannian volume forms (alternative generally covariant integration measure densities) on the space–time manifold are studied in some detail. These models involve an additional

Virasoro symmetry of constrained KP hierarchies

R^2

Darboux-Bäcklund solutions of SL(p, q) KP-KdV hierarchies, constrained generalized Toda lattices, and two-matrix string model

R2 (square of the scalar curvature) term as well as scalar matter field potentials of appropriate form so that the pertinent action is invariant under global Weyl-scale symmetry. Scale invariance is spontaneously broken upon integration of the equations of motion for the auxiliary volume-form degrees of freedom. After performing transition to the physical Einstein frame we obtain: (1) an effective potential for the scalar field with two flat regions which allows for a unified description of both early universe inflation as well as of present dark energy epoch; (2) for a definite parameter range the model possesses a non-singular “emergent universe” solution which describes an initial phase of evolution that precedes the inflationary phase; (3) for a reasonable choice of the parameters the present model conforms to the Planck Collaboration data.

Unified dark energy and dust dark matter dual to quadratic purely kinetic K-essence

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.

Symplectic actions on coadjoint orbits

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.