doi:10.1155/2008/792632 Research Article The Finite Discrete KP Hierarchy and the Rational Functions

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.


Introduction
Few decades ago, an unexpected relation between the control theory and the integrable systems 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 Rat Λ n .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.

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 Λ be the n by n-matrix with ones the matrix entry equals to 1 in the first upper diagonal and zero in the remaining entries and Λ T its transpose.The matrix Λ is like a shift operator of coordinates for vectors in R n .The Λ k matrix, 2 ≤ k ≤ n − 1, is a zero matrix except in the kth upper diagonal where it has ones.Note that Λ n Λ T n O, where O is the zero matrix.
Let L be a matrix where D i are diagonal matrices.The entries of L are assumed to be functions depending on parameters t t 1 , . . ., t n−1 .
Definition 2.1.The finite discrete KP hierarchy is the Lax system where L ≥ L > denotes the strictly upper triangular part of L, analogously L ≤ L < denotes the strictly lower triangular part of L.
Now, let us assume that for an operator defined as in 2.2 , one can find an invertible matrix S: where in 2.4 and 2.6 , S i and W i are diagonal matrices, then L will be of Lax type.
From now, we can only consider solutions of 2.3 which are of the Lax type.The operator S 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 Λ.Note that L Λ is the most simple solution of 2.3 and for it S I.
If there is a dressing operator such that where L SΛS −1 , then L is of the Lax type, and moreover it satisfies 2.3 .Conversely, if L is a solution of 2.3 which is of the Lax type, then there exists a dressing operator S of L, such that S is solution of 2.7 .This operator S 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 n, it is always possible to find matrices L 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 M admits a Gauss-Borel decomposition if M can be written in the form where M ≤ has ones on the principal diagonal and M ≥ has nonzero elements on the principal diagonal.Decomposition 2.9 is equivalent to where G 0 is a diagonal matrix with nonzero elements.It can be proved that the Gauss-Borel decomposition is unique.M admits the Gauss-Borel decomposition if and only if M k / 0 , k 1, . . ., n, where M k is the determinant of the k-order principal submatrix.Of particular interest will be the matrix space M * of matrices M depending on t, admitting a Gauss-Borel decomposition and for which M ≥ 0 I, where I is the identity matrix.Let Z be the formal 1-form of L given by 12 which is equivalent to the Zakharov-Shabat equations, where by definition

2.13
Let which is a trivial solution of 2.12 .Each solution of 2.3 yields a solution of the linear equation in M * and conversely for a solution of 2.15 in M * we can build a matrix L which will be a solution of 2.3 .The solutions of 2.15 take the form

2.16
Let us consider the Gauss-Borel factorization of U : where S −1 is a lower triangular matrix with ones on the principal diagonal and Y is an upper triangular matrix with nonzero elements on the principal diagonal.
For U ∈ M * we have that where S −1 0 is a matrix, which takes the form of 2.4 .It is easy to show that if U ∈ M * , then in 2.17 S is a Sato-Wilson operator, therefore using the Gauss-Borel factorization of U and doing L SΛS −1 , we obtain a solution of 2.3 for which

Rational functions induced by the finite discrete KP hierarchy solutions
Let rat p n be the space of rational functions of grade n and fixed denominator p z z n p n−1 z n−1 • • • p 0 .It is possible to see that an element f z ∈ rat p n always admits a unique factorization this matrix L p will be denominated by p-operator note that L p is of the form 2.2 .Observe that the characteristic polynomials of L p is equal to the given p z .At this point, it is convenient to inspect the Lax operator as function of its dressing operators.Let us see it for n 2 and n 3:

3.3
From these simple examples, it follows that the unique p-operator that is also a Lax operator is justly L p Λ.This assertion holds also for any arbitrary dimension of matrix L p and it shows that rat n its definition is given next is the unique space that could be characterized by this hierarchy, using the Gauss-Borel factorization.
Let rat n be a space of rational functions f z ∈ rat p n such that with a n / 0 and a i ∈ R for any i.On this space, it is usual to introduce the equivalence relation We denote the equivalence class to which belongs f z and by f and the set rat n \∼ { f : f rat n } by Rat Λ n .As mentioned before, any element of rat n can be written in the form where C T 0 1, 0, . . ., 0 , B T 0 a 1 , a 2 , . . ., a n are constant vectors in R n , and a i are the coefficients of f z .
Note that The functions f rat n expressed according to 3.6 can be considered as the transfer functions of the linear dynamical system dx dτ Λx τ B 0 u τ , y τ C T 0 x τ .

3.8
Proposition 3.1.The linear dynamical system 3.8 associated to f z C T 0 zI − Λ −1 B 0 rat n is controllable and observable. Proof.Indeed, if a n / 0 .This fact implicates the controllability of 3.8 .Also is the canonical base on R n .This fact implies that 3.8 is observable.Proposition 3.2.Let U ∈ M * , U S −1 Y and B, C be defined as follows: 12 then B and C satisfy the linear equations where L SΛS −1 .
Proof.We have that and since Having in mind that S is a Sato-Wilson operator and ∂U/∂t k Λ k U,

3.18
On the other hand, we have that for k 1, . . ., n − 1.Since S verifies 2.7 , it is Sato-Wilson matrix then

3.20
From 3.19 and 3.20 , its follows that Let us consider the function

3.30
Since U I 0 I, then

3.31
We see that F z, t 1 , . . ., t n−1 rat n .From 3.30 , we have that As U I is an upper triangular matrix with ones on the diagonal, then the independent terms of q n−1 and of the numerator of f z coincide.
We can characterize the flow of 3.30 .Taking derivate respect to t k , we obtain

3.34
The flows on rat n determined by L I have the form where V defines the numerator of F and it has the form

3.36
Let us notice that 3.35 is similar to the flows considered by Brockett and Faybusovich in 1 .
Let L I t , B t , C t be a triple, where L I t is the above fixed solution of 2.3 ; B t and C t are solutions of 3.13 .This triple gives a flow of 3.30 on rat n defined by the initial conditions L I 0 Λ, B 0 B 0 , C 0 C 0 that determine the triple Λ, B 0 , C 0 .In such way, we have the nontrivial flow Λ, B 0 , C 0 → L I , B, C on R 2n n 2 .The equivalence relation 3.5 induces an equivalence relation ≺ on the set of triples Λ, B 0 , C 0 such that Λ, B 1 0 , C 0 ≺ Λ, B 2 0 , C 0 , if B 1 0 αB 2 0 , α R \{0}.

3.37
Definition 3.5.The set of solutions L I t , B t , C t of compatible systems 2.3 and 3.13 for any equivalence class Λ, B 0 , C 0 that satisfies 3.6 is denominated by the moduli space of systems 2.3 and 3.13 .We denote this moduli space as M Λ .Now, we will discuss the correspondence between Rat Λ n and M Λ and we will built a one-to-one mapping from Rat Λ n to M Λ .
We can obtain a flow on Rat Λ n by means of 3.35 , determined by f z in the following way:

1 0T C 0 . As S − 1 0T
that the initial conditions hold.Since B Y B 0 and Y 0 I, then B 0 B 0 .On other hand, due to C S −1 T C 0 , we have that C 0 S −1 0 T C 0 S −is an upper triangular matrix with ones on the diagonal, then S −1 0 T C 0 C 0 .
C and B satisfy 3.13 , L I satisfy 2.3 , and L I 0 Λ.Thus any f Rat Λ n defines a solution L I , B, C determined by the initial conditions Λ, B 0 , C 0 .So we have a mapping β : Rat Λ n −→ M Λ .3.39 Conversely, any solution L I , B, C ∈ M Λ induces a flow 3.30 on Rat Λ n which originates into f z at t 0. Furthermore, by 3.31 we obtain the class Λ, B 0 , C 0 .Thus β is an onto mapping form Rat Λ n to M Λ .Theorem 3.6.The mapping β defined in 3.39 is a bijection.