On generalized Lax equation of the Lax triple of KP Hierarchy

In terms of the operator Nambu 3-bracket and the Lax pair (L, B_n) of the KP hierarchy, we propose the generalized Lax equation with respect to the Lax triple (L, B_n, B_m). The intriguing results are that we derive the KP equation and another integrable equation in the KP hierarchy from the generalized Lax equation with the different Lax triples (L, B_n, B_m). Furthermore we derive some no integrable evolution equations and present their single soliton solutions.     

The quasiclassical limit of the symmetry constraint of the KP hierarchy and the dispersionless KP hierarchy with self-consistent sources

Abstract

For the first time we show that the quasiclassical limit of the symmetry constraint of the Sato operator for the KP hierarchy leads to the generalized Zakharov reduction of the Sato function for the dispersionless KP (dKP) hierarchy which has been proved to be result of symmetry constraint of the dKP hierarchy recently. By either regarding the symmetry constrained dKP hierarchy as its stationary case or taking the dispersionless limit of the KP hierarchy with self-consistent sources directly, we construct a new integrable dispersionless hierarchy, i.e., the dKP hierarchy with self-consistent sources and find its associated conservation equations (or equations of Hamilton-Jacobi type). Some solutions of the dKP equation with self-consistent sources are also obtained by hodograph transformations.     

An extended integrable fractional-order KP soliton hierarchy

In this Letter, we consider the modified derivatives and integrals of fractional-order pseudo-differential operators. A sequence of Lax KP equations hierarchy and extended fractional KP (fKP) hierarchy are introduced, and the fKP hierarchy has Lax presentations with the extended Lax operators. In the case of the extension with the half-order pseudo-differential operators, a new integrable fKP hierarchy is obtained. A few particular examples of fractional order will be listed, together with their Lax pairs.     

Canonical property of the Miura maps between the MKP and KP hierarchies,continuous and discrete

Apart from the case of the KP hierarchy, all known Miura maps between integrable Hamiltonian systems had been proven to be canonical. The remaining KP case is settled below. As a corollary, it is shown that the KP hierarchy is a factor — hierarchy of the mKP one, with the kernel consisting of a single scalar field. A discrete mKP hierarchy and the associated Miura map are constructed, and the latter is shown to be canonical as well. As in the continuous case, this implies that one can extend the discrete KP hierarchy by a single new field into an extended discrete KP hierarchy in such a way that the extended discrete Miura map mKPeKP is a canonical isomorphism.     

Lattice equations,hierarchies and hamiltonian structures: The Kadomtsev-Petviashvili equation

By applying a suitable continuum limit to a lattice version of the Kadomtsev-Petviashvili (KP) equation we obtain an infinite hierarchy of integrable partial differential-difference equations defined on a two-dimensional lattice, together with the associated hamiltonian structure.     

(4,3) and (5,2) Nonlinear Evolution Equations of (2+1)-Dimensional KP Hierarchy

The Kadometsev-Petviashvili (KP) equation is generalized to a class of equations of the (2+1)-dimensional KP hierarchy. The (4,3) and (5,2) integrable evolution equations with high nonlinearity are given in detail.     

Quantum fields presentation and generating functions of symplectic Schur functions and symplectic universal characters

This paper is concerned with construction of quantum fields presentation and generating functions of symplectic Schur functions and symplectic universal characters. The boson-fermion correspondence for these symmetric functions have been presented. In virtue of quantum fields, we derive a series of infinite order nonlinear integrable equations, namely, universal character hierarchy, symplectic KP hierarchy and symplectic universal character hierarchy, respectively. In addition, the solutions of these integrable systems have been discussed.     

Some quasi-periodic solutions to the Kadometsev-Petviashvili and modified Kadometsev-Petviashvili equations

The Kadometsev-Petviashvili (KP) and modified KP (mKP) equations are retrieved from the first two soliton equations of coupled Korteweg-de Vries (cKdV) hierarchy. Based on the nonlinearization of Lax pairs, the KP and mKP equations are ultimately reduced to integrable finite-dimensional Hamiltonian systems in view of the r-matrix theory. Finally, the resulting Hamiltonian flows are linearized in Abel-Jacobi coordinates, such that some specially explicit quasi-periodic solutions to the KP and mKP equations are synchronously given in terms of theta functions through the Jacobi inversion.     

On the Geometry of Darboux Transformations for the KP Hierarchy and its Connection with the Discrete KP Hierarchy

We tackle the problem of interpreting the Darboux transformation for the KP hierarchy and its relations with the modified KP hierarchy from a geometric point of view. This is achieved by introducing the concept of a Darboux covering. We construct a Darboux covering of the KP equations and obtain a new hierarchy of equations, which we call the Darboux-KP hierarchy (DKP). We employ the DKP equations to discuss the relationships among the KP equations, the modified KP equations, and the discrete KP equations. Our approach also handles the various reductions of the KP hierarchy. We show that the KP hierarchy is a projection of the DKP, the mKP hierarchy is a DKP restriction to a suitable invariant submanifold, and that the discrete KP equations are obtained as iterations of the DKP ones. Received: 23 July 1996 / Accepted: 6 January 1997     

Virasoro symmetry of constrained KP hierarchies

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-Bäcklund orbits of initial purely differential Lax operators. Our construction establishes the condition for commutativity of the additional-symmetry flows with the discrete Darboux-Bäcklund transformations of cKP hierarchies leading to a new derivation of the string-equation constraint in matrix models.     

Symmetries of the Classical Integrable Systems and 2-Dimensional Quantum Gravity: a ‘Map’

Abstract

We draw attention to the connections recently established by others between the classical integrable KdV and KP hierarchies in 1 + 1 and 2 + 1 dimensions respectively and the matrix models which relate to the partition functions of 2-dimensional (1 + 1 dimensional) quantum gravity. The symmetries of the classical KP hierarchy in 2 + 1 dimensions are fundamental to this connection.     

Hodge Integrals and Integrable Hierarchies

We show that the generating series of some Hodge integrals involving one or two partitions are τ-functions of the KP hierarchy or the 2-Toda hierarchy, respectively. We also reformulate the results as a relationship between the relative invariants of some local Calabi-Yau geometries and integrable hierarchies and present some more examples using the topological vertex.     

Möbius invariant integrable lattice equations associated with KP and 2DTL hierarchies

The integrable lattice equations arising in the context of singular manifold equations for scalar, multicomponent KP hierarchies and 2D Toda lattice hierarchy are considered. They generate the corresponding continuous hierarchy of singular manifold equations, its Bäcklund transformations and different forms of superposition principles; their distinctive feature is invariance under the action of Möbius transformation. Geometric interpretation of these discrete equations is given.     

Addition Formulae of Discrete KP,q-KP and Two-Component BKP Systems

In this paper,we construct the addition formulae for several integrable hierarchies,including the discrete KP,the q-deformed KP,the two-component BKP and the D type Drinfeld–Sokolov hierarchies.With the help of the Hirota bilinear equations and τ functions of different kinds of KP hierarchies,we prove that these addition formulae are equivalent to these hierarchies.These studies show that the addition formula in the research of the integrable systems has good universality.     

A new “dual” symmetry structure of the KP hierarchy

A new infinite set of commuting additional (“ghost”) symmetries is proposed for the KP-type integrable hierarchy. These symmetries allow for a Lax representation in which they are realized as standard isospectral flows. This gives rise to a new double-KP hierarchy embedding “ghost” and original KP-type Lax hierarchies connected to each other via a “duality” mapping exchanging the isospectral and “ghost” “times”. A new representation of the 2D Toda lattice hierarchy as a special Darboux-Bäcklund orbit of the double-KP hierarchy is found and parametrized entirely in terms of (adjoint) eigenfunctions of the original KP subsystem.     

Symmetry constraints for dispersionless integrable equations and systems of hydrodynamic type

Symmetry constraints for (2+1)-dimensional dispersionless integrable equations are considered. It is demonstrated that they naturally lead to certain systems of hydrodynamic type which arise within the reduction method. One also easily obtains an associated complex curve (Sato function) and corresponding generating equations. Dispersionless KP and 2DTL hierarchy are considered as illustrative examples.     

Bilinear Identities and Hirota's Bilinear Forms for an Extended Kadomtsev-Petviashvili Hierarchy

Abstract

In this paper, we construct the bilinear identities for the wave functions of an extended Kadomtsev-Petviashvili (KP) hierarchy, which is the KP hierarchy with particular extended flows. By introducing an auxiliary parameter, whose flow corresponds to the so-called squared eigenfunction symmetry of KP hierarchy, we find the tau-function for this extended KP hierarchy. It is shown that the bilinear identities will generate all the Hirota's bilinear equations for the zero-curvature forms of the extended KP hierarchy, which includes two types of KP equation with self-consistent sources (KPSCS). The Hirota's bilinear equations obtained in this paper for the KPSCS are in different forms by comparing with the existing results.     

Self-Consistent Sources for Integrable Equations Via Deformations of Binary Darboux Transformations

We reveal the origin and structure of self-consistent source extensions of integrable equations from the perspective of binary Darboux transformations. They arise via a deformation of the potential that is central in this method. As examples, we obtain in particular matrix versions of self-consistent source extensions of the KdV, Boussinesq, sine-Gordon, nonlinear Schrödinger, KP, Davey–Stewartson, two-dimensional Toda lattice and discrete KP equation. We also recover a (2+1)-dimensional version of the Yajima–Oikawa system from a deformation of the pKP hierarchy. By construction, these systems are accompanied by a hetero binary Darboux transformation, which generates solutions of such a system from a solution of the source-free system and additionally solutions of an associated linear system and its adjoint. The essence of all this is encoded in universal equations in the framework of bidifferential calculus.     

Properties of supersymmetric integrable systems of KP type

The recently proposed supersymmetric extensions of reduced Kadomtsev-Petviashvili (KP) integrable hierarchies in N = 1, 2 superspace are shown to contain in the purely bosonic limit new types of ordinary non-supersymmetric integrable systems. The latter are coupled systems of several multi-component non-linear Schr?dinger-like hierarchies whose basic nonlinear evolution equations contain additional quintic and higher-derivative nonlinear terms. Also, we obtain the N = 2 supersymmetric extension of Toda chain model as Darboux-B?cklund orbit of the simplest reduced N = 2 super-KP hierarchy and find its explicit solution. Received 13 September 2001 Published online 2 October 2002 RID="a" ID="a"e-mail: nissimov@inrne.bas.bg RID="b" ID="b"e-mail: svetlana@inrne.bas.bg