Discrete KP Hierarchy Research Articles

Constrained discrete KP hierarchy: The constraint on the tau functions and gauge transformations

Constrained discrete KP hierarchy: The constraint on the tau functions and gauge transformations

Minimal realizations and scaling invariance of the discrete KP hierarchy and its strict version

Minimal realizations and scaling invariance of the discrete KP hierarchy and its strict version

Several Integrable Properties of the Discrete KP Hierarchy Obtained by Means of Gauge Transformations

The gauge transformation is a powerful method to solve integrable hierarchies. In the discrete KP hierarchy, there are two basic gauge transformations, T D ( q ) = Λ( q ) ∘ Δ ∘ q −1 and T I ( r ) = Λ −1 ( r −1 ) ∘ Δ −1 ∘ r. This paper is concerned with several integrable properties of the discrete KP hierarchy found by means of gauge transformations. The successive applications of T I to the constrained discrete KP hierarchy have been presented. Meanwhile, we derive two novel discrete equations as the orbits of gauge transformations T D and T I for the constrained KP hierarchy. Moreover, with the help of the spectral representations and the gauge transformations, Fay-similar equations of the tau functions for the constrained discrete KP have been investigated. In addition, we also discuss the relation between the gauge transformation and an additional symmetry for the discrete KP hierarchy.

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.

Extensions of the discrete KP hierarchy and its strict version

Доказано, что и дискретная иерархия Кадомцева-Петвиашвили, и ее строгая версия могут быть расширены до более широкого класса деформаций, которые удовлетворяют более общему набору уравнений Лакса. Доказано, что обе расширенные иерархии обладают соответствующими линеаризациями, позволяющими геометрически построить решения этих расширенных иерархий.

Matrix-valued Laurent polynomials, parametric linear systems and integrable systems

In this paper, we study transfer functions corresponding to parametric linear systems whose coefficients are block matrices. Thus, these transfer functions constitute Laurent polynomials whose coefficients are square matrices. We assume that block matrices defining the parametric linear systems are solutions of an integrable hierarchy called by us, the block matrices version of the finite discrete KP hierarchy, which is introduced and studied with certain detail in this paper. We see that the linear system defined of the simplest solution of the integrable system is controllable and observable. Then, as a consequence of this fact, it is possible to verify that any solution of the integrable hierarchy, obtained by the dressing method of the simplest solution, defines a parametric linear system which is also controllable and observable.

Strict Versions of Integrable Hierarchies in Pseudodifference Operators and the Related Cauchy Problems

In the algebra PsΔ of pseudodifference operators, we consider two deformations of the Lie subalgebra spanned by positive powers of an invertible constant first-degree pseudodifference operator Λ0. The first deformation is by the group in PsΔ corresponding to the Lie subalgebra Ps<0 of elements of negative degree, and the second is by the group corresponding to the Lie subalgebra PsΔ≤0 of elements of degree zero or lower. We require that the evolution equations of both deformations be certain compatible Lax equations that are determined by choosing a Lie subalgebra depending on Λ0 that respectively complements the Lie subalgebra PsΔ<0 or PsΔ≤0. This yields two integrable hierarchies associated with Λ0, where the hierarchy of the wider deformation is called the strict version of the first because of the form of the Lax equations. For Λ0 equal to the matrix of the shift operator, the hierarchy corresponding to the simplest deformation is called the discrete KP hierarchy. We show that the two hierarchies have an equivalent zero-curvature form and conclude by discussing the solvability of the related Cauchy problems.

A discrete hierarchy of double bracket equations and a class of negative power series

The space of negative power series of $z$ on $\{z\in \mathbb{C}:|z|&gt;1\}$ can also be parametrized by means of a system of double bracket differential equations. To show this parametrization we introduce a group factorization for equation system. This work, for the case of a double bracket system, is a continuation of an earlier study discussed in The discrete KP hierarchy and the negative power series on the complex plane. Comp. and App. Math. 32 (2013), 483-493 for the case of one bracket system.

Matrix integral solutions to the discrete KP hierarchy and its Pfaffianized version

Matrix integrals used in random matrix theory for the study of eigenvalues of Hermitian ensembles have been shown to provide τ-functions for several hierarchies of integrable equations. In this article, we extend this relation by showing that such integrals can also provide τ-functions for the discrete KP hierarchy and a coupled version of the same hierarchy obtained through the process of Pfaffianization. To do so, we consider the first equation of the discrete KP hierarchy, the Hirota–Miwa equation. We write the Wronskian determinant solutions to the Hirota–Miwa equation and consider a particular form of matrix integrals, which we show is an example of those Wronskian solutions. The argument is then generalized to the whole hierarchy. A similar strategy is used for the Pfaffianized version of the hierarchy except that in that case, the solutions are written in terms of Pfaffians rather than determinants.

Virasoro symmetry of the constrained multicomponent Kadomtsev–Petviashvili hierarchy and its integrable discretization

In this paper, we construct the Virasoro type additional symmetries of a kind of constrained multi-component KP hierarchy and give the Virasoro flow equation on eigenfunctions and adjoint eigenfunctions. It can also be seen that the algebraic structure of the Virasoro symmetry is kept after discretization from the constrained multi-component KP hierarchy to the discrete constrained multi-component KP hierarchy.

Ghost symmetry of the discrete KP hierarchy

In this paper, with the help of the S function and ghost symmetry for the discrete KP hierarchy which is a semi-discrete version of the KP hierarchy, the ghost flow on its eigenfunction (adjoint eigenfunction) and the spectral representation of its Baker–Akhiezer function and adjoint Baker–Akhiezer function are derived. From these observations above, some important distinctions between the discrete KP hierarchy and KP hierarchy are shown. Also we give the ghost flow on the tau function and another kind of proof of the ASvM formula of the discrete KP hierarchy.

Solutions to the ultradiscrete KP hierarchy and its reductions

We propose a recursive representation of solutions to an ultradiscrete analogue of the discrete KP hierarchy, which is the master equation of discrete soliton equations. We also propose a class of solutions which can be used to start the recursion. Finally, as an application, we discuss the compatibility condition of ultradiscrete soliton equations.

The Virasoro action on the tau function for the constrained discrete KP hierarchy

With the help of the squared eigenfunction potential, the action of the Virasoro symmetry on the tau function of the constrained discrete KP hierarchy is derived.

The discrete KP hierarchy and the negative power series on the complex plane

In this article, we have connected the control theory of linear infinite-dimensional systems and the discrete KP integrable system. We show how the space of negative power series of \(z\) on \(\{z\in \mathbb{C }: |z| > 1 \}\) can be parametrized by means of an integrable system. This study is a kind of extension to infinite-dimensional case of some results discussed in Felipe and Lopez-Reyes (Discret Dyn Nat Soc 2008, 2008).

Solutions of the bigraded Toda hierarchy

The (N, M)-bigraded Toda hierarchy is an extension of the original Toda lattice hierarchy. The pair of numbers (N, M) represents the band structure of the Lax matrix which has N upper and M lower diagonals, and the original one is referred to as the (1, 1)-bigraded Toda hierarchy. Because of this band structure, one can introduce M + N − 1 commuting flows which give a parametrization of a small phase space for a topological field theory. In this paper, first we show that there exists a natural symmetry between the (N, M)- and (M, N)-bigraded Toda hierarchies. We then derive the Hirota bilinear form for those commuting flows, which consist of two-dimensional Toda hierarchy, the discrete KP hierarchy and its Bäcklund transformations. We also discuss the solution structure of the (N, M)-bigraded Toda equation in terms of the moment matrix defined via the wave operators associated with the Lax operator and construct some of the explicit solutions. In particular, we give the rational solutions which are expressed by the products of the Schur polynomials corresponding to the non-rectangular Young diagrams.

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.

Determinant solutions of the nonautonomous discrete Toda equation associated with the deautonomized discrete KP hierarchy

It is shown that the nonautonomous discrete Toda equation and its Backlund transformation can be derived from the reduction of the hierarchy of the discrete KP equation and the discrete two-dimensional Toda equation. Some explicit examples of the determinant solutions of the nonautonomous discrete Toda equation including the Askey-Wilson polynomial are presented. Finally we discuss the relationship between the nonautonomous discrete Toda system and the nonautonomous discrete Lotka-Volterra equation.

A new extended discrete KP hierarchy and a generalized dressing method

Inspired by the squared eigenfunction symmetry constraint, we introduce a new τk-flow by ‘extending’ a specific tn-flow of a discrete KP hierarchy (DKPH). We construct an extended discrete KPH (exDKPH), which consists of tn-flow, τk-flow and tn evolution of eigenfunction and adjoint eigenfunctions, and its Lax representation. The exDKPH contains two types of discrete KP equation with self-consistent sources (DKPESCS). Two reductions of exDKPH are obtained. The generalized dressing approach for solving the exDKPH is proposed and the N-soliton solutions of two types of the DKPESCS are presented.