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.     

Recursion Operators for Dispersionless KP Hierarchy

Based on the corresponding theorem between dispersionless KP(dKP)hierarchy and -dependent KP ( KP) hierarchy, a general formal representation of the recursion operators for dKP hierarchy under n-reduction is given in a systematical way from the corresponding KP hierarchy. To illustrate this method, the recursion operators for dKP hierarchy under 2-reduction and 3-reduction are calculated in detail.     

Recursion Operators for Dispersionless KP Hierarchy

Based on the corresponding theorem between dispersionless KP (dKP) hierarchy and h-dependent KP (hKP) hierarchy, a general formal representation of the recursion operators for dKP hierarchy under n-reduction is given in a systematical way from the corresponding hKP hierarchy. To illustrate this method, the recursion operators for dKP hierarchy under 2-reduction and 3-reduction are calculated in detail.     

Volume Preserving Multidimensional Integrable Systems and Nambu–Poisson Geometry

Abstract

In this paper we study generalized classes of volume preserving multidimensional integrable systems via Nambu–Poisson mechanics. These integrable systems belong to the same class of dispersionless KP type equation. Hence they bear a close resemblance to the self dual Einstein equation. All these dispersionless KP and dToda type equations can be studied via twistor geometry, by using the method of Gindikin’s pencil of two forms. Following this approach we study the twistor construction of our volume preserving systems.     

Coisotropic Deformations of Associative Algebras and Dispersionless Integrable Hierarchies

The paper is an inquiry of the algebraic foundations of the theory of dispersionless integrable hierarchies, like the dispersionless KP and modified KP hierarchies and the universal Whitham hierarchy of genus zero. It stands out for the idea of interpreting these hierarchies as equations of coisotropic deformations for the structure constants of certain associative algebras. It discusses the link between the structure constants and Hirota’s tau function, and shows that the dispersionless Hirota bilinear equations are, within this approach, a way of writing the associativity conditions for the structure constants in terms of the tau function. It also suggests a simple interpretation of the algebro-geometric construction of the universal Whitham equations of genus zero due to Krichever.     

Vertex Operator Solutions to the Discrete KP-Hierarchy

Vertex operators, which are disguised Darboux maps, transform solutions of the KP equation into new ones. In this paper, we show that the bi-infinite sequence obtained by Darboux transforming an arbitrary KP solution recursively forward and backwards, yields a solution to the discrete KP-hierarchy. The latter is a KP hierarchy where the continuous space $x$-variable gets replaced by a discrete $n$-variable. The fact that these sequences satisfy the discrete KP hierarchy is tantamount to certain bilinear relations connecting the consecutive KP solutions in the sequence. At the Grassmannian level, these relations are equivalent to a very simple fact, which is the nesting of the associated infinite-dimensional planes (flag). The discrete KP hierarchy can thus be viewed as a container for an entire ensemble of vertex or Darboux generated KP solutions. It turns out that many new and old systems lead to such discrete (semi-infinite) solutions, like sequences of soliton solutions, with more and more solitons, sequences of Calogero–Moser systems, having more and more particles, just to mention a few examples; this is developed in [3]. In this paper, as another example, we show that the q-KP hierarchy maps, via a kind of Fourier transform, into the discrete KP hierarchy, enabling us to write down a very large class of solutions to the q-KP hierarchy. This was also reported in a brief note [4]. Received: 27 August 1998 / Accepted: 24 November 1998     

Hamiltonian Structures of Reductions of the Benney System

We show how to construct the Hamiltonian structures of any reduction of the Benney chain (dispersionless KP). The construction follows the scheme suggested by Ferapontov, leading in general to nonlocal Hamiltonian structures. In some special cases these reduce to local structures. All the geometric objects which define the Poisson bracket, the metric, connection and Riemann curvature, are obtained explicitly, in terms of the n-parameter family of conformal maps associated with the reduction.     

Dispersionless noncommutative Kadomtsev–Petviashvili hierarchy: Multi-Hamiltonian structure and large N-like V-algebras

We propose a dispersionless limit of the NC Kadomtsev–Petviashvili hierarchy. Multi-Hamiltonian formulation, l-reductions and relevant W-algebras are investigated. As examples, Gelfand–Dickey (GD) Poisson brackets for dispersionless NCKdV and NC Boussinesq hierarchies are constructed explicitly. The associated GD Poisson algebras are shown large N-like analogs of classical V-algebras.     

The Construction of Frobenius Manifolds¶from KP tau-Functions

Frobenius manifolds (solutions of WDVV equations) in canonical coordinates are determined by the system of Darboux–Egoroff equations. This system of partial differential equations appears as a specific subset of the n-component KP hierarchy. KP representation theory and the related Sato infinite Grassmannian are used to construct solutions of this Darboux–Egoroff system and the related Frobenius manifolds. Finally we show that for these solutions Dubrovin's isomonodromy tau-function can be expressed in the KP tau-function. Received: 1 September 1998 / Accepted: 7 March 1999     

A Note on the Dispersionless Dym Hierarchy

We discuss the Miura map as well as the Poisson algebras associated with the dispersionless Dym hierarchy. Particularly, we study explicitly the bi-Hamiltonian structure of a truncated Dym system with two variables, in which a new hierarchy flow generated by logarithmic Hamiltonians appears. We then show that this new hierarchy emerges naturally from the topological recursion relation in the Landau–Ginzburg formulation.     

Analytic Functions and Integrable Hierarchies–Characterization of Tau Functions

We prove the dispersionless Hirota equations for the dispersionless Toda, dispersionless coupled modified KP and dispersionless KP hierarchies using an idea from classical complex analysis. We also prove that the Hirota equations characterize the tau functions for each of these hierarchies. As a result, we establish the links between the hierarchies.     

Topological Landau-Ginzburg theory with a rational potential and the dispersionless KP hierarchy

Based on the dispersionless KP (dKP) theory, we study a topological Landau-Ginzburg (LG) theory characterized by a rational potential. Writing the dKP hierarchy in a general form treating all the primaries in an equal basis, we find that the hierarchy naturally includes the dispersionless (continuous) limit of Toda hierarchy and its generalizations having a finite number of primaries. Several flat solutions of the topological LG theory are obtained in this formulation, and are identified with those discussed by Dubrovin. We explicitly construct gravitational descendants for all the primary fields. Giving a residue formulae for the 3-point functions of the fields, we show that these 3-point functions satisfy the topological recursion relation. The string equation is obtained as the generalized hodograph solutions of the dKP hierarchy, which show that all the gravitational effects to the constitutive equations (2-point functions) can be renormalized into the coupling constants in the small phase space. Supported in part by NSF grant DMS-9403597.     

Higher dimensional classicalW-algebras

ClassicalW-algebras in higher dimensions are constructed. This is achieved by generalizing the classical Gel'fand-Dickey brackets to the commutative limit of the ring of classical pseudodifferential operators in arbitrary dimension. TheseW-algebras are the Poisson structures associated with a higher dimensional version of the Khokhlov-Zabolotskaya hierarchy (dispersionless KP-hierarchy). The two dimensional case is worked out explicitly and it is shown that the role of DiffS(1) is taken by the algebra of generators of local diffeomorphisms in two dimensions.     

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.     

Bihamiltonian Cohomology of KdV Brackets

Using spectral sequences techniques we compute the bihamiltonian cohomology groups of the pencil of Poisson brackets of dispersionless KdV hierarchy. In particular, this proves a conjecture of Liu and Zhang about the vanishing of such cohomology groups.     

Quasi-classical limit of Toda hierarchy andW-infinity symmetries

Previous results on quasi-classical limit of the KP hierarchy and itsW-infinity symmetries are extended to the Toda hierarchy. The Planck constant now emerges as the spacing unit of difference operators in the Lax formalism. Basic notions, such as dressing operators, Baker-Akhiezer functions, and tau function, are redefined.W _{1 +} symmetries of the Toda hierarchy are realized by suitable rescaling of the Date-Jimbo-Kashiara-Miwa vertex operators. These symmetries are contracted tow _{1 +} symmetries of the dispersionless hierarchy through their action on the tau function.     

Combinatorics of Dispersionless Integrable Systems and Universality in Random Matrix Theory

It is well-known that the partition function of the unitary ensembles of random matrices is given by a τ-function of the Toda lattice hierarchy and those of the orthogonal and symplectic ensembles are τ-functions of the Pfaff lattice hierarchy. In these cases the asymptotic expansions of the free energies given by the logarithm of the partition functions lead to the dispersionless (i.e. continuous) limits for the Toda and Pfaff lattice hierarchies. There is a universality between all three ensembles of random matrices, one consequence of which is that the leading orders of the free energy for large matrices agree. In this paper, this universality, in the case of Gaussian ensembles, is explicitly demonstrated by computing the leading orders of the free energies in the expansions. We also show that the free energy as the solution of the dispersionless Toda lattice hierarchy gives a solution of the dispersionless Pfaff lattice hierarchy, which implies that this universality holds in general for the leading orders of the unitary, orthogonal, and symplectic ensembles. We also find an explicit formula for the two point function F _nm which represents the number of connected ribbon graphs with two vertices of degrees n and m on a sphere. The derivation is based on the Faber polynomials defined on the spectral curve of the dispersionless Toda lattice hierarchy, and \frac1nmF_nm{\frac{1}{nm}F_{nm}} are the Grunsky coefficients of the Faber polynomials.     

Logarithmic Deformations of the Rational Superpotential/Landau-Ginzburg Construction of Solutions of the WDVV Equations

The superpotential in the Landau-Ginzburg construction of solutions to the Witten-Dijkgraaf-Verlinde-Verlinde (or WDVV) equations is modified to include logarithmic terms. This results in deformations - quadratic in the deformation parameters- of the normal prepotential solutions of the WDVV equations. Such solutions satisfy various pseudo-quasi-homogeneity conditions, on assigning a notional weight to the deformation parameters. These solutions originate in the so-called ‘water-bag’ reductions of the dispersionless KP hierarchy. This construction includes, as a special case, deformations which are polynomial in the flat coordinates, resulting in a new class of polynomial solutions of the WDVV equations.