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Discrete Dynamics in Nature and Society
Volume 2008, Article ID 792632, 10 pages
http://dx.doi.org/10.1155/2008/792632
Research Article

The Finite Discrete KP Hierarchy and the Rational Functions

1CIMAT, 36000 Guanajuato, Mexico and ICIMAF, 10600 La Habana, Cuba
2Departamento de Matemática, Facultad de Ciencias Exactas y Naturales, Universidad de Antioquia, Medellín 1226, Colombia

Received 27 November 2007; Accepted 6 June 2008

Academic Editor: Stevo Stević

Copyright © 2008 Raúl Felipe and Nancy López. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Abstract

The set of all rational functions with any fixed denominator that simultaneously nullify in the infinite point is parametrized by means of a well-known integrable system: a finite dimensional version of the discrete KP hierarchy. This type of study was originated in Y. Nakamura's works who used others integrable systems. Our work proves that the finite discrete KP hierarchy completely parametrizes the space of rational functions of the form , where is a polynomial of order with nonzero independent coefficent. More exactly, it is proved that there exists a bijection from to the moduli space of solutions of the finite discrete KP hierarchy and a compatible linear system.

1. Introduction

Few decades ago, an unexpected relation between the control theory and the integrable sys- tems was revealed. Papers [1, 2] which deal with problems related to those discussed here are examples of these researches.

A large part of this research activity shows that some nonlinear integrable systems have rich information about the moduli space of certain classes of solutions of linear dynamical systems. In particular, they have relation with spaces of certain classes of rational functions. Also according to the state space realization theory, some rational functions can be associated with a controllable and observable linear dynamical system.

A convenient property of these spaces of rational functions is that they can be considered as varieties, thus a question arises about the study of its moduli space. Maybe, in this context, the fundamental role of the integrable systems is its compatibility with families of controllable and observable linear dynamical systems.

During the last ten years, the subject of integrable system has been enriched in a remarkable way by its extensions to the other setting, notably, to discrete case of the KP hierarchy. These developments have originated in the mathematical physics. In the new settings, many of the classical tools are available, for example, we point out one of them which is basic to this paper, the Gauss-Borel decomposition for the discrete KP hierarchy, proposed by Felipe and Ongay [3] as an extension of the Mulase’s algebraic geometric approach to the KP hierarchy [4]. This decomposition allows us to consider on an almost equal footing the cases of semi-infinite and bi-infinite matrices.

In this paper, we use the well-known theory of the discrete KP hierarchy studied, for instance, in [3], restricted to finite matrices to characterize . Thus we will give another example of nonlinear integrable system that also has the property of completely parameterizing some kind of rational function space.

We must observe that an interesting property of the finite discrete KP hierarchy is that it contains the full Kostant-Toda equation.

2. An Algebraic Geometric Approach for the Finite Discrete KP Hierarchy

The goals of this section are as follows.

(1)The first goal is to introduce a natural commuting finite hierarchy of flows. We make three comments of this hierarchy. First, it can be defined by a Lax-type operator (matrix) with respect to the shift matrix and its transpose. Second, the Lax matrix introduced admits in certain cases a dressing matrix, in terms of which the hierarchy can be rewritten. Third, there is a Sato-Wilson matrix “to dress” the shift matrix. We mention that the situation is similar to the Sato theory and his dressing technique (pseudodifferential theory).(2)The second is to review the integrability in the sense of Frobenius for the hierarchy introduced which turns out very simple in this context: the key point in our method is the so-called Gauss-Borel decomposition. It also verifies that the finite discrete KP hierarchy like any integrable system is always related to some kind of group factorization.

Next, we describe the corresponding Mulase’s approach associated to the finite discrete KP hierarchy as it is considered in [3]. We omit the proofs because in [3] these were given in a more general background.

Let be the by -matrix with ones (the matrix entry equals to 1) in the first upper diagonal and zero in the remaining entriesand its transpose. The matrix is like a shift operator of coordinates for vectors in The matrix, , is a zero matrix except in the th upper diagonal where it has ones. Note that where is the zero matrix.

Let be a matrixwhere are diagonal matrices. The entries of are assumed to be functions depending on parameters .

Definition 2.1. The finite discrete KP hierarchy is the Lax systemwhere denotes the (strictly) upper triangular part of , analogously denotes the (strictly) lower triangular part of .

Now, let us assume that for an operator defined as in (2.2), one can find an invertible matrix :such thatwhere in (2.4) and (2.6), and are diagonal matrices, then will be of Lax type.

From now, we can only consider solutions of (2.3) which are of the Lax type. The operator is called a dressing operator, and the decomposition (2.5) is unique, up to right multiplication by an invertible matrix, taking the form of (2.4) that commutes with . Note that is the most simple solution of (2.3) and for it .

If there is a dressing operator such thatwhere then is of the Lax type, and moreover it satisfies (2.3). Conversely, if is a solution of (2.3) which is of the Lax type, then there exists a dressing operator of , such that is solution of (2.7). This operator is called the Sato-Wilson matrix (see [3] for more details).

In this point, it is very important to observe that for any given order , it is always possible to find matrices of the form (2.2) which are not of Lax type. It is explained bellow with more details.

It was shown in [3] that from (2.3) the Zakharov-Shabat equations follow

We say that a matrix admits a Gauss-Borel decomposition if can be written in the formwhere has ones on the principal diagonal and has nonzero elements on the principal diagonal. Decomposition (2.9) is equivalent towhere is a diagonal matrix with nonzero elements. It can be proved that the Gauss-Borel decomposition is unique. admits the Gauss-Borel decomposition if and only if , where is the determinant of the -order principal submatrix. Of particular interest will be the matrix space of matrices depending on , admitting a Gauss-Borel decomposition and for which , where is the identity matrix.

Let be the formal -form of given by

If satisfies (2.3), then satisfieswhich is equivalent to the Zakharov-Shabat equations, where by definition

Letwhich is a trivial solution of (2.12). Each solution of (2.3) yields a solution of the linear equationin and conversely for a solution of (2.15) in we can build a matrix which will be a solution of (2.3). The solutions of (2.15) take the form

Let us consider the Gauss-Borel factorization of : where is a lower triangular matrix with ones on the principal diagonal and is an upper triangular matrix with nonzero elements on the principal diagonal.

For we have thatthenwhere is a matrix, which takes the form of (2.4).

It is easy to show that if , then in (2.17) is a Sato-Wilson operator, therefore using the Gauss-Borel factorization of and doing , we obtain a solution of (2.3) for which (see [3]).

3. Rational Functions Induced by the Finite Discrete KP Hierarchy Solutions

Let be the space of rational functions of grade and fixed denominator . It is possible to see that an element always admits a unique factorizationwhere , andthis matrix will be denominated by -operator (note that is of the form (2.2)). Observe that the characteristic polynomials of is equal to the given .

At this point, it is convenient to inspect the Lax operator as function of its dressing operators. Let us see it for and :

From these simple examples, it follows that the unique -operator that is also a Lax operator is justly = . This assertion holds also for any arbitrary dimension of matrix and it shows that (its definition is given next) is the unique space that could be characterized by this hierarchy, using the Gauss-Borel factorization.

Let be a space of rational functions such thatwith and for any . On this space, it is usual to introduce the equivalence relation

We denote the equivalence class to which belongs and by and the set by . As mentioned before, any element of can be written in the formwhere are constant vectors in , and are the coef- ficients of .

Note that

The functions expressed according to (3.6) can be considered as the transfer functions of the linear dynamical system

Proposition 3.1. The linear dynamical system (3.8) associated to is controllable and observable.

Proof. Indeed,becauseif . This fact implicates the controllability of (3.8). Alsobecause , is the canonical base on . This fact implies that (3.8) is observable.

Proposition 3.2. Let , and , be defined as follows:then and satisfy the linear equations where

Proof. We have thatfor Being , we obtainand since
Having in mind that is a Sato-Wilson operator and ,then
On the other hand, we have thatfor Since verifies (2.7), it is Sato-Wilson matrix then
From (3.19) and (3.20), its follows that
Let us show that the initial conditions hold.
Since and , then . On other hand, due to we have that . As is an upper triangular matrix with ones on the diagonal, then .

Remark 3.3. It is interesting to note that independently of the selection of as initial condition in the factorization of , the flow for always begins in

We restrict ourselves to consider a solution of (2.3) to which and we denote it by . In this case, we must take by (2.20).

Next, let us define a linear dynamical system of parameter using , and in the following way

Note that for , we obtain (3.8).

Proposition 3.4. The linear dynamical system (3.22) is controllable and observable.

Proof. Indeed,
Since and commute, then
By (3.9) and due to , we have thatfrom wherethen (3.22) is controllable.
Now we will show the observability of (3.32).
By (3.11) and since we havefrom where

Let us consider the function

Since , then

We see that . From (3.30), we have that

As is an upper triangular matrix with ones on the diagonal, then the independent terms of and of the numerator of coincide.

We can characterize the flow of (3.30). Taking derivate respect to we obtaindoing , we have

The flows on determined by have the formwhere defines the numerator of and it has the form

Let us notice that (3.35) is similar to the flows considered by Brockett and Faybusovich in [1].

Let be a triple, where is the above fixed solution of (2.3); and are solutions of (3.13). This triple gives a flow of (3.30) on defined by the initial conditions that determine the triple .

In such way, we have the nontrivial flow on .

The equivalence relation (3.5) induces an equivalence relation on the set of triples such that

Definition 3.5. The set of solutions of compatible systems (2.3) and (3.13) for any equivalence class that satisfies (3.6) is denominated by the moduli space of systems (2.3) and (3.13). We denote this moduli space as .

Now, we will discuss the correspondence between and and we will built a one-to-one mapping from to

We can obtain a flow on by means of (3.35), determined by in the following way:where and satisfy (3.13), satisfy (2.3), and . Thus any defines a solution determined by the initial conditions . So we have a mapping

Conversely, any solution induces a flow (3.30) on which originates into at . Furthermore, by (3.31) we obtain the class . Thus is an onto mapping form to .

Theorem 3.6. The mapping defined in (3.39) is a bijection.

Acknowledgments

The first author was supported in part under CONACYT grant 37558E, and in part under the Cuba National Project “Theory and algorithms for the solution of problems in algebra and geometry” and the second author was partially supported by COLCIENCIAS project 395 and CODI project 9889-E01251.

References

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