# The Finite Discrete KP Hierarchy  and the Rational Functions

Authors: Raúl Felipe; Nancy López
Journal: Discrete Dynamics in Nature and Society
 (2008)

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## 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 spaceRatΛ(n) of rational functions of the form f(x)=q(x)/zn, where q(x) is a polynomial of order n−1 with nonzero independent coefficent. More exactly, it is proved that there exists a bijection from RatΛ(n) to the moduli space of solutions of the finite discrete KP hierarchy and a compatible linear system.


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## Body

## 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 RatΛ(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.


## 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.LetΛ 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(2.1)Λ=(01⋯⋯⋯00010⋯0⋮⋱⋱⋱⋱⋮⋮⋱⋱⋱⋱000⋯⋯0100⋯⋯00)and ΛT its transpose. The matrix Λ is like a shift operator of coordinates for
vectors in Rn. The Λk matrix, 2≤k≤n−1, is a zero matrix except in the kth upper diagonal where it has
ones. Note that Λn=ΛTn=O, where O is the zero matrix.LetL be a matrix(2.2)L=Λ+D0+∑i=1,…,n−1Di(ΛT)i,where Di are diagonal matrices. The entries of L are assumed to be functions depending on
parameters t=(t1,…,tn−1).Definition 2.1.
The finite discrete KP hierarchy is the Lax system(2.3)∂L∂tk=[L≥k,L],k=1,…,n−1,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:(2.4)S=I+S1ΛT+S2(ΛT)2+⋯+Sn−1(ΛT)n−1such that(2.5)L=SΛS−1,(2.6)S−1=I+W1ΛT+W2(ΛT)2+⋯+Wn−1(ΛT)n−1where in (2.4) and (2.6), Si and Wi 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(2.7)∂S∂tk=−L<kS,k=1,…,n−1,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 ordern,
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 (2.8)∂L≥i∂tj−∂L≥j∂ti=[L≥i,L≥j],i≠j,⁢i,⁢j=1,…,n−1.We say that a
matrixM admits a Gauss-Borel decomposition if M can be written in the form(2.9)M=M≤M≥,where M≤ has ones on the
principal diagonal and M≥ has nonzero elements on the principal
diagonal. Decomposition (2.9) is equivalent to(2.10)M=(I+G<G0−1)(I+G0+G>),where G0 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 Mk≠0,k=1,…,n,
where Mk 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.LetZ be the formal 1-form of L given by(2.11)Z=∑k=1n−1L≥kdtk.IfL satisfies (2.3), then Z satisfies(2.12)dZ=12[Z,Z]which is equivalent to the
Zakharov-Shabat equations, where by definition(2.13)[Z,Z]=∑i,j=1n−1[L≥i,L≥j]dti⁢dtj.Let(2.14)Ω=∑k=1n−1Λkdtkwhich is a trivial solution of
(2.12).
Each solution of (2.3) yields a solution of the linear equation(2.15)dU=ΩUin 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)U=e∑k=1n−1ΛktkU0.Let us consider
the Gauss-Borel factorization ofU:(2.17)U=S−1Y, 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.ForU∈M∗ we have that(2.18)U0=U(0)=S−1(0)Y(0)=S−1(0)I=S0−1,then(2.19)U=e∑k=1n−1ΛktkU0=e∑k=1n−1ΛktkS0−1,where S0−1 is a matrix, which takes the form of (2.4).It is easy to
show that ifU∈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 (2.20)L(0)=S0ΛS0−1=L0,(see [3]).


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

Letratp(n) be the space of rational functions of grade n and fixed denominator p(z)=zn+pn−1zn−1+⋯+p0.
It is possible to see  that an element f(z)∈ratp(n) always admits a unique
factorization(3.1)f(z)=q(z)p(z)=C0T(zIn−Lp)−1B0,where C0T=(1,0,…),B0T=(bn−1⋅bn−2,…,b0),q(z)=bn−1zn−1+bn−2zn−2+⋯+b0, and(3.2)Lp=(−pn−110⋯0−pn−20⋱⋱⋮⋮⋮⋱⋱0−p10⋯⋱1−p00⋯⋯0),this matrix Lp will be denominated by p-operator (note that Lp is of the form
(2.2)). Observe that the characteristic polynomials of Lp 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 forn=2 and n=3: (3.3)n=2n=3S−1(10a1)(100a10cb1)S(10−a1)(100−a10ab−c−b1)L(−a1−a2a)(a10−a2+c−a+b1(a2−c)b−ac(ab−c)−b2−b).From these simple examples, it follows that the uniquep-operator that is also a Lax operator is
justly Lp = Λ.
This assertion holds also for any arbitrary dimension of matrix Lp 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.Letrat(n) be a space of rational functions f(z)∈ratp(n) such that(3.4)f(z)=a1zn−1+a2zn−2+⋯+anzn=q(z)znwith an≠0 and ai∈ℝ for any i.
On this space, it is usual to introduce the
equivalence relation(3.5)f∼αfαϵR∖{0}.We denote  the equivalence class to which belongsf(z) and by [f] and the set rat(n)∖∼={[f]:fϵrat(n)} by RatΛ(n). As mentioned before, any element of rat(n) can be written in the form(3.6)f(z)=C0T(zI−Λ)−1B0,where C0T=(1,0,…,0),B0T=(a1,a2,…,an) are constant vectors in Rn,
and ai are the coef- ficients of f(z).Note
that(3.7)αf(z)=C0T(zI−Λ)−1(αB0).The functionsfϵrat(n) expressed according to (3.6) can be considered
as the transfer functions of the linear dynamical
system(3.8)dxdτ=Λx(τ)+B0u(τ),y(τ)=C0Tx(τ).Proposition 3.1.
The linear dynamical system
(3.8) associated to f(z)=C0T(zI−Λ)−1B0ϵrat(n) is controllable and observable.Proof.
Indeed,(3.9)rang((B0)(ΛB0)⋯(Λn−1B0))=nbecause(3.10)|a1a2ana2a30.....⋯....an−1an0an00|=−(ann)≠0if an≠0.
This fact implicates the controllability of (3.8). Also(3.11)rang((C0)(ΛTC0)⋯(ΛTn−1C0))=nbecause (ΛT)iC0,i=0,…,n−1,
is the canonical base on Rn.
This fact implies that (3.8) is observable.Proposition 3.2.
LetU∈M∗, U=S−1Y and B, C be defined as
follows:(3.12)B(t1,…,tn−1)=Y(t1,…,tn−1)B0,C(t1,…,tn−1)=(S−1(t1,…,tn−1))TC0,then B and C satisfy the linear equations (3.13)∂B∂tk=L≥kB,∂C∂tk=(L<k)TC,⁢k=1,…,n−1,B(0)=B0,⁢C(0)=C0,where L=SΛS−1.Proof.
We have that(3.14)∂B∂tk=∂Y∂tkB0for k=1,…,n−1. Being U=S−1Y,
we obtain(3.15)∂U∂tk=∂S−1∂tkY+S−1∂Y∂tkand since(3.16)S−1S=I∂S−1∂tkS+S−1∂S∂tk=0,∂U∂tk=−S−1∂S∂tkS−1Y+S−1∂Y∂tk.
Having in mind
thatS is a Sato-Wilson operator and ∂U/∂tk=ΛkU,(3.17)ΛkU=∂U∂tk=−S−1∂S∂tkS−1Y+S−1∂Y∂tk,SΛkU+∂S∂tkS−1Y=∂Y∂tk,(SΛkS−1)Y−L<kY=∂Y∂tk,L≥kY=(Lk−L<k)Y=∂Y∂tk,then(3.18)∂B∂tk=∂Y∂tkB0=L≥kYB0=L≥kB.
On the other
hand, we have that(3.19)∂C∂tk=∂(S−1)T∂tkB0,C=(S−1)TC0=(ST)−1C0for k=1,…,n−1. Since S verifies (2.7), it is Sato-Wilson matrix
then(3.20)∂ST∂tk=−STL<k.
From (3.19) and
(3.20), its follows that(3.21)∂C∂tk=∂(S−1)T∂tkC0=∂(ST)−1∂tkC0=(L<k)T(ST)−1C0=(L<k)TC.
Let us show
that the initial conditions hold.
SinceB=YB0 and Y(0)=I,
then B(0)=B0.
On other hand, due to C=(S−1)TC0, we have that C(0)=(S−1(0))TC0=(S0−1)TC0.
As (S0−1)T is an upper triangular matrix with
ones on the diagonal, then (S0−1)TC0=C0.Remark 3.3.
It is interesting to note that independently
of the selection ofS0−1 as initial condition in the factorization of U,
the flow for C always begins in C0.We restrict
ourselves to consider a solutionL of (2.3) to which LI(0)=Λ and we denote it by LI.
In this case, we must take S0=I by (2.20).Next, let us
define a linear dynamical system of parametert=(t1,…,tn−1) using B(t), C(t), and LI(t) in the following way(3.22)ddτx(τ,t)=L(τ,t)x(τ,t)+B(τ,t)u(τ,t),y(τ,t1)=CT(τ,t)x(τ,t).Note that fort1=⋯=tn−1=0, we obtain (3.8).Proposition 3.4.
The linear dynamical system
(3.22) is controllable and observable.Proof.
Indeed,(3.23)rang((B)(LB)⋯(Ln−1B))=rang((YB0)(SΛS−1YB0)⋯(SΛn−1S−1YB0))=rang(S((UB0)(ΛUB0)⋯(Λn−1UB0))).
SinceΛ and U commute, then(3.24)rang((B)(LB)⋯(Ln−1B))=rang(SU(B0)(ΛB0)⋯(Λn−1B0))=rang(Y(B0)(ΛB0)⋯(Λn−1B0)).
By (3.9) and due
to det⁡Y≠0,
we have that(3.25)rang(Y(B0)(ΛB0)⋯(Λn−1B0))=nfrom where(3.26)rang((B)(LB)⋯(Ln−1B))=nthen (3.22) is controllable.
Now we will show the observability of (3.32).(3.27)rang((C)(LC)⋯(Ln−1C))=rang(((ST)−1C0)((ST)−1ΛTST(ST)−1C0)⋯((ST)−1(ΛT)n−1ST(ST)−1C0))=rang((ST)−1(C0(ΛTC0)⋯((ΛT)n−1C0)).
By (3.11) and
since det⁡(ST)−1≠0, we have(3.28)rang((ST)−1(C0(ΛTC0)⋯((ΛT)n−1C0))=nfrom where(3.29)rang((C)(LC)⋯(Ln−1C))=n.Let us consider
the function(3.30)F(z,t1,…,tn−1)=CT(zI−LI)−1B=CT(S(zI−Λ)S−1)−1B=CTS(zI−Λ)−1S−1B=((S−1)TC0)TS(zI−Λ)−1S−1B=C0TS−1S(zI−Λ)−1S−1YB0=C0T(zI−Λ)−1UIB0.SinceUI(0)=I,  then(3.31)F(z,0)=f(z)=C0T(zI−LI(0))−1B0.We see thatF(z,t1,…,tn−1)ϵrat(n).
From (3.30), we have that(3.32)F(z,t1,…,tn−1)=b1zn−1+b2zn−2+⋯+bnzn=qn−1(z,t1,…,tn−1)zn.AsUI is an upper triangular matrix with
ones on the diagonal, then the independent terms of qn−1 and of the numerator of f(z) coincide.We can
characterize the flow of (3.30). Taking derivate respect to tk, we obtain(3.33)∂F∂tk=C0T(zI−LI)−1∂UI∂tkB0,k=1,…,n−1,doing V=UIB0,
we have(3.34)F=C0T(zI−Λ)−1V.The flows onrat(n) determined by LI have the form(3.35)∂V∂tk=ΛkV,k=1,…,n−1,where V defines the numerator of F and it has the form(3.36)V=e∑k=1n−1ΛktkB0.Let us notice
that (3.35) is similar to the flows considered by Brockett and Faybusovich in
[1].Let(LI(t),B(t),C(t)) be a triple, where LI(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 LI(0)=Λ,B(0)=B0,⁢C(0)=C0 that determine the triple (Λ,B0,C0).In such way, we
have the nontrivial flow(Λ,B0,C0)→(LI,B,C) on R2n+n2.The equivalence
relation (3.5) induces an equivalence relation ≺ on the set of triples (Λ,B0,C0) such that(3.37)(Λ,B01,C0)≺(Λ,B02,C0),ifB01=αB02,αϵR∖{0}.Definition 3.5.
The set of solutions(LI(t),B(t),C(t)) of compatible systems (2.3) and (3.13) for any
equivalence class [(Λ,B0,C0)] that satisfies (3.6) is denominated
by the moduli space of systems (2.3) and (3.13). We
denote this moduli space as MΛ.Now, we will
discuss the correspondence betweenRatΛ(n) and MΛ and we will built a one-to-one mapping from RatΛ(n) to MΛ.We can obtain a
flow onRatΛ(n) by means of (3.35), determined by f(z) in the following way:(3.38)F(z,t1,…,tn−1)=C0T(zI−Λ)−1V=C0T(zI−Λ)−1e∑k=1n−1ΛktkB0=C0T(zI−Λ)−1S−1YB0=((S−1)TC0)T(zI−LI)−1YB0=CT(zI−LI)−1B,where C and B satisfy (3.13), LI satisfy (2.3), and LI(0)=Λ.
Thus any [f]ϵRatΛ(n) defines a solution (LI,B,C) determined by the initial conditions [(Λ,B0,C0)].
So we have a mapping(3.39)β:RatΛ(n)→MΛ.Conversely, any
solution(LI,B,C)∈MΛ induces a flow (3.30) on RatΛ(n) which originates into f(z) at t=0.
Furthermore, by (3.31) we obtain the class [(Λ,B0,C0)].
Thus β is an onto mapping form RatΛ(n) to MΛ.Theorem 3.6.
The mappingβ defined in (3.39) is a bijection.



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