dKP Hierarchy Research Articles

W algebra and eigenvalues of the low-order string constraint operators for the discrete KP hierarchy

Firstly, for any p ∈ N , the string constraint operators without the redundant variables are calculated, which are imposed by the string equation on the τ -function of the p -reduced discrete Kadomtsev–Petviashvili (dKP) hierarchy. Furthermore, the commutators of the string constraint operators without the redundant variables are calculated. It is found that although the string constraint operators without the redundant variables are different from the original string constraint operators with the redundant variables, they also form a Virasoro algebra and a W algebra after a transformation. Moreover, using the algebraic structure, eigenvalues of the low-order string equation constraint operators are calculated. Based on the obtained eigenvalues, it is also proved that the τ -function of the p -reduced dKP hierarchy constrained by the string equation is a vacuum vector for a Witt algebra. When p = 2 , it is coincident with the classical fact that the τ -function of the Korteweg–de Vries hierarchy constrained by the string equation is a vacuum vector for a Virasoro algebra.

Extensions of the discrete KP hierarchy and its strict version

Abstract: We show that both the dKP hierarchy and its strict version can be extended to a wider class of deformations satisfying a larger set of Lax equations. We prove that both extended hierarchies have appropriate linearizations allowing a geometric construction of their solutions.

Dispersionless integrable hierarchies andGL(2, ℝ) geometry

AbstractParaconformal orGL(2, ℝ) geometry on ann-dimensional manifoldMis defined by a field of rational normal curves of degreen– 1 in the projectivised cotangent bundle ℙT*M. Such geometry is known to arise on solution spaces of ODEs with vanishing Wünschmann (Doubrov–Wilczynski) invariants. In this paper we discuss yet another natural source ofGL(2, ℝ) structures, namely dispersionless integrable hierarchies of PDEs such as the dispersionless Kadomtsev–Petviashvili (dKP) hierarchy. In the latter context,GL(2, ℝ) structures coincide with the characteristic variety (principal symbol) of the hierarchy.Dispersionless hierarchies provide explicit examples of particularly interesting classes of involutiveGL(2, ℝ) structures studied in the literature. Thus, we obtain torsion-freeGL(2, ℝ) structures of Bryant [5] that appeared in the context of exotic holonomy in dimension four, as well as totally geodesicGL(2, ℝ) structures of Krynski [33]. The latter possess a compatible affine connection (with torsion) and a two-parameter family of totally geodesicα-manifolds (coming from the dispersionless Lax equations), which makes them a natural generalisation of the Einstein–Weyl geometry.Our main result states that involutiveGL(2, ℝ) structures are governed by a dispersionless integrable system whose general local solution depends on 2n– 4 arbitrary functions of 3 variables. This establishes integrability of the system of Wünschmann conditions.

Polynomial tau-functions of BKP and DKP hierarchies

We construct all polynomial tau-functions of the BKP, DKP, and MDKP hierarches.

A 3-Lie algebra and the dKP Hierarchy

In terms of a 3-Lie algebra and the classical Poisson bracket {Bn,L} of the dKP hierarchy, a special 3-bracket {Bm,Bn,L} is proposed. When m = 0 or m = 1, the 3-lax equation \\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$${{\\partial L} \\over {\\partial t}} = \\{ {B_m},{B_n},L)$$\\end{document} is the dKP hierarchy and the corresponding proof is given. Meanwhile, for the generalized case (m,n), the generalized dKP hierarchy is also investigated.

Miura and auto-Bäcklund transformations for the discrete KP and mKP hierarchies and their constrained cases

In this paper, the Miura and auto-Bäcklund transformations for the dKP and mdKP hierarchies and their constrained cases are investigated. The combination of the Miura and reverse-Miura transformations are used to derive the auto-Bäcklund transformations. Also some applications of above results are considered.

A geometric construction of solutions of the strict dKP(Λ0) hierarchy

In a previous paper we associated to each invertible constant pseudo difference operator Λ 0 of degree one, two integrable hierarchies in the algebra of pseudo difference operators Ps Δ , the so-called dKP( Λ 0 ) hierarchy and its strict version. We show here first that both hierarchies can be described as the compatibility conditions for a proper linearization. Next we present a geometric framework for the construction of solutions of the hierarchies, i.e. we associate to each hierarchy an infinite dimensional variety such that to each point of the variety one can construct a solution of the corresponding hierarchy. This yields a Segal–Wilson type framework for all these integrable hierarchies.

Multi-variable reductions of the dispersionless DKP hierarchy

We consider multi-variable reductions of the dispersionless DKP hierarchy (the dispersionless limit of the Pfaff lattice) in the elliptic parametrization. The reduction is given by a system of elliptic Löwner equations supplemented by a system of partial differential equations of hydrodynamic type. The compatibility conditions for the elliptic Löwner equations are derived. They are elliptic analogues of the Gibbons–Tsarev equations. We prove solvability of the hydrodynamic type system by means of the generalized hodograph method. The associated diagonal metric is proved to be of the Egorov type.

The partition function of the Bures ensemble as the τ-function of BKP and DKP hierarchies: continuous and discrete

The relationship between matrix integrals and integrable systems was revealed more than 20 years ago. As is known, matrix integrals over a Gaussian ensemble used in random matrix theory could act as the τ-function of several hierarchies of integrable systems. In this article, we will show that the time-dependent partition function of the Bures ensemble, whose measure has many interesting geometric properties, could act as the τ-function of BKP and DKP hierarchies. In addition, if discrete time variables are introduced, then this partition function could act as the τ-function of discrete BKP and DKP hierarchies. In particular, there are some links between the partition function of the Bures ensemble and Toda-type equations.

Dispersionless DKP hierarchy and the elliptic Löwner equation

We show that the dispersionless DKP hierarchy (the dispersionless limit of the Pfaff lattice) admits a suggestive reformulation through elliptic functions. We also consider one-variable reductions of the dispersionless DKP hierarchy and show that they are described by an elliptic version of the Löwner equation. With a particular choice of the driving function, the latter appears to be closely related to the Painlevé VI equation with a special choice of parameters.

The (n,1)-Reduced DKP Hierarchy, the String Equation and W Constraints

The total descendent potential of a simple singularity satisfies the Kac-Wakimoto principal hierarchy. Bakalov and Milanov showed recently that it is also a highest weight vector for the corresponding W-algebra. This was used by Liu, Yang and Zhang to prove its uniqueness. We construct this principal hierarchy of type D in a different way, viz. as a reduction of some DKP hierarchy. This gives a Lax type and a Grassmannian formulation of this hierarchy. We show in particular that the string equation induces a large part of the W constraints of Bakalov and Milanov. These constraints are not only given on the tau function, but also in terms of the Lax and Orlov-Schulman operators.

Cohomological and Poisson structures and integrable hierarchies in tautological subbundles for Birkhoff strata of the Sato Grassmannian

We consider cohomological and Poisson structures associated with the special tautological subbundles \(TB_{W_{1,2, \ldots n} }\) for the Birkhoff strata of the Sato Grassmannian. We show that the tangent bundles of \(TB_{W_{1,2, \ldots n} }\) are isomorphic to the linear spaces of two-coboundaries with vanishing Harrison cohomology modules. A special class of two-coboundaries is provided by a system of integrable quasilinear partial differential equations. For the big cell, it is the hierarchy of dispersionless Kadomtsev-Petvishvili (dKP) equations. We also demonstrate that the families of ideals for algebraic varieties in \(TB_{W_{1,2, \ldots n} }\) can be viewed as Poisson ideals. This observation establishes a relation between families of algebraic curves in \(TB_{W_{\hat S} }\) and coisotropic deformations of such curves of zero and nonzero genus described by hierarchies of systems of hydrodynamic type; the dKP hierarchy is such a hierarchy. We note the interrelation between cohomological and Poisson structures.

The string equation and the τ-function Witt constraints for the discrete Kadomtsev-Petviashvili hierarchy

Using the tool of pseudo-difference operators, the string equation for the discrete Kadomtsev-Petviashvili (KP) hierarchy is introduced and a general expression for the string equation is formulated. Furthermore, the operators of the constraints that the string equation imposes on the τ-function of the p-reduced dKP hierarchy are presented. The algebra, which part of the constraints spans, and eigenvalues of part of the constraints corresponding to the τ-function are calculated. It is also shown that this algebra is exactly a Witt algebra not only for p = 2, but also for a general \documentclass[12pt]{minimal}\begin{document}$p\in \mathbb {N}$\end{document}p∈N, and that the τ-function of the p-reduced dKP hierarchy constrained by the string equation is a vacuum vector of a Witt algebra.

Recursion Operators for Dispersionless KP Hierarchy

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

Frobenius Manifold for the Dispersionless Kadomtsev-Petviashvili Equation

We consider a Frobenius structure associated with the dispersionless Kadomtsev – Petviashvili equation. This is done, essentially, by applying a continuous analogue of the finite dimensional theory in the space of Schwartz functions on the line. The potential of the Frobenius manifold is found to be a logarithmic energy with quadratic external field. Following the construction of the principal hierarchy, we construct a set of infinitely many commuting flows, which extends the classical dKP hierarchy.

Non-Hamiltonian generalizations of the dispersionless 2DTL hierarchy

We consider two-component integrable generalizations of the dispersionless two-dimensional Toda lattice (2DTL) hierarchy connected with non-Hamiltonian vector fields, similar to the Manakov–Santini hierarchy generalizing the dKP hierarchy. They form a one-parametric family connected by hodograph-type transformations. Generating equations and Lax–Sato equations are introduced, and a dressing scheme based on the vector nonlinear Riemann problem is formulated. The simplest two-component generalization of the dispersionless 2DTL equation is derived, and its differential reduction analogous to the Dunajski interpolating system is presented. A symmetric two-component generalization of the dispersionless elliptic 2DTL equation is also constructed.

The determinant representation of the gauge transformation for the discrete KP hierarchy

A successive gauge transformation operator Tn+k for the discrete KP (dKP) hierarchy is defined, which is involved with two types of gauge transformations operators. The determinant representation of the Tn+k is established and it is used to get a new τ function τΔ(n+k) 4 of the dKP hierarchy from an initial τΔ. In this process, we introduce a generalized discrete Wronskian determinant and some useful properties of discrete difference operators.

Sato's Bäcklund transformations, additional symmetries and ASvM formula for the discrete KP hierarchy

Two kinds of symmetries, Sato's Bäcklund transformations and additional symmetries, for the discrete KP (dKP) hierarchy are introduced, and the ASvM formula which demonstrates the equivalence of these two kinds of symmetries is obtained. In this process the Fay identity and the difference Fay identity of the dKP hierarchy are introduced and the ASvM formula in the form of tau function is calculated.

On a class of reductions of the Manakov–Santini hierarchy connected with the interpolating system

Using the Lax–Sato formulation of the Manakov–Santini hierarchy, we introduce a class of reductions such that the zero-order reduction of this class corresponds to the dKP hierarchy, and the first-order reduction gives the hierarchy associated with the interpolating system introduced by Dunajski. We present the Lax–Sato form of a reduced hierarchy for the interpolating system and also for the reduction of arbitrary order. Similar to the dKP hierarchy, the Lax–Sato equations for L (the Lax function) split from the Lax–Sato equations for M (the Orlov function) due to the reduction, and the reduced hierarchy for an arbitrary order of reduction is defined by Lax–Sato equations for L only. A characterization of the class of reductions in terms of the dressing data is given. We also consider a waterbag reduction of the interpolating system hierarchy, which defines (1+1)-dimensional systems of hydrodynamic type.

Integrable GL(2) geometry and hydrodynamic partial differential equations

This article is a local analysis of integrable GL(2)-structures of degree 4. A GL(2)-structure of degree n corresponds to a distribution of rational normal cones over a manifold M of dimension (n+1). Integrability corresponds to the existence of many submanifolds that are spanned by lines in the cones. These GL(2)-structures are important because they naturally arise from a certain family of second-order hyperbolic PDEs in three variables that are integrable via hydrodynamic reduction. Familiar examples include the wave equation, the first flow of the dKP equation, and the Boyer--Finley equation. The main results are a structure theorem for integrable GL(2)-structures, a classification for connected integrable GL(2)-structures, and an equivalence between local integrable GL(2)-structures and Hessian hydrodynamic hyperbolic PDEs in three variables. This yields natural geometric characterizations of the wave equation, the first flow of the dKP hierarchy, and several others. It also provides an intrinsic, coordinate-free infrastructure to describe a large class of hydrodynamic integrable systems in three variables.