An Elementary Approach to the Polynomial -functions

Ref. SISSA 117/98/FM

An Elementary Approach to the Polynomial  -functions of the KP Hierarchy 1

2

Gregorio Falqui , Franco Magri ,

3

4

Marco Pedroni , and Jorge P. Zubelli 1

2

3

SISSA, Via Beirut 2/4, I-34014 Trieste, Italy E{mail: [email protected]

Dipartimento di Matematica, Universita di Milano Via C. Saldini 50, I-20133 Milano, Italy E{mail: [email protected] Dipartimento di Matematica, Universita di Genova Via Dodecaneso 35, I-16146 Genova, Italy E{mail: [email protected]

IMPA { CNPq Est. D. Castorina 110, Rio de Janeiro, RJ 22460, Brazil E{mail: [email protected] 4

Abstract

We construct in an elementary and simple way the solutions of the KP hierarchy associated with polynomial  -functions. Our starting point is a geometric approach to soliton equations, which relies on the concept of a bihamiltonian system. As a consequence, we establish a Wronskian formula for the polynomial  -functions of the KP hierarchy. This formula, though known in the literature, is obtained in a very direct way.

During the course of this work, GF, FM, and MP were sponsored by the Italian M. U. R. S. T. and by the Italian National Research Council (CNR) through the GNFM. JPZ was sponsored by the Brazilian National Research Council (CNPq) through grant MA 521329/94-9.

1 Introduction Among the many remarkable properties of soliton equations one nds the existence of rational (with respect to the space variable x) solutions. It was discussed for the Korteweg-de Vries (KdV) equation as well as for the Kadomtsev-Petviashvili (KP) equation in many places, e.g., [1, 2, 4, 7, 13, 14, 16]. It was also observed that the poles of these solutions move according to nite-dimensional integrable Hamiltonian systems such as the Calogero-Moser hierarchy (see also the more recent papers [21] and [26]). Then, in the context of the Sato theory of soliton equations [20, 18], rational solutions of the KP hierarchy were seen as the solutions associated to polynomial  -functions. These solutions are rational with respect to all times of the hierarchy, and represent particular instances of rational solutions of the KP equation. The class of KP solutions associated to polynomial  -functions has been shown [27, 25] to yield bispectral rings of dierential operators. The bispectral property has been recently subject to intense research (see, e.g., [3]). This concept, which originated in certain investigations of A. Grunbaum concerning computerized tomography, goes as follows: a dierential operator L(x; d=dx) is called bispectral if it possesses a family of eigenfunctions (x; z) that is also a family of eigenfunctions for some dierential operator A(z; d=dz). It is rather remarkable that one of the lemmas that was instrumental in providing a proof of bispectrality for the wave functions associated to polynomial  -function in [27] was also useful here (see Corollary 5.5). In this paper we recover the solutions of the KP hierarchy associated with polynomial  -functions in an elementary way using the framework introduced in [11], where a geometric approach to the linearization of the KP equations is presented. It starts from the geometry of bihamiltonian manifolds and arrives at a system of in nite-matrix Riccati equations, which are linearized by means of elementary methods. These equations already appeared in [23], where they are shown to describe linear ows on the Sato Grassmannian. For this reason, the above-mentioned system was called the Sato System in [11]. Here we explicitly determine the solutions of the Sato System when the initial condition has only a nite number of nonzero entries, and we show that these solutions give rise to the solutions of the KP hierarchy associated with polynomial  -functions. One can nd in the literature several constructions of these solutions. They make use of nowadays standard but non trivial tools such as e.g., pseudodierential operators and in nite-dimensional Grassmannians, and allow to obtain a wider class of solutions. Our construction has the advantage of being to some extent self-contained, and likely can be generalized in order to obtain other solutions than the ones considered in this paper. For example, if the initial data is in the class of Hilbert-Schmidt operators, it is not hard to see that the solution remains in such class when evolved by the ows of the Sato Systems. It would then be natural to try extending some of the results presented herein by considering more general classes of initial data and their corresponding  -functions. 1

The plan for this article is the following: In Section 2 we recall how to pass from the KP hierarchy (in conservation-law form) to the Sato System, through another important object, which is the Central System. In Section 3 we review the linearization procedure for the Sato System and we add some information on the solution of the linearized system for arbitrary initial data. In Section 4 the initial condition is assumed to vanish outside the m  n upper (left) corner. Under those assumptions the Sato System can be explicitly solved, and the solutions can be expressed in terms of a polynomial  -function, as shown in Section 5. Finally, an example is presented in Section 6.

2 The KP hierarchy, the Central System, and the Sato System In this section (and at the beginning of the next one) we will recall some results from [11], where a description of the linearization of the KP hierarchy is given starting from the bihamiltonian approach to the KdV hierarchy. We refer to that paper for motivations, details, and proofs. First of all we introduce the KP equations in a form which, although equivalent, is P dierent from the usual one | see, however, [6, 24]. Let h(x; z) = z + i hi (x)z i be a Laurent series whose coecients hi belong to a given space F of C 1-functions. Let us consider the Faa di Bruno iterates of h, de ned as 1

hj

( +1)

= @xh j + hh j ; ( )

h = 1:

( )

(2.1)

(0)

They span a subspace H := hh j ij which is complementary to H := hzj ij< . Let  and  be the corresponding projections, and let us call ( )

+

0

0

H k :=  (zk ) ( )

+

(2.2)

+

the KP currents. They are the unique Laurent series in H having the asymptotic expansion H k = zk + O(z ). +

( )

1

De nition 2.1 The KP equations are the equations @h = @ H k @tk x

( )

for the Laurent series h.

(2.3)

We refer to [11] for the relation with the usual description of the KP equations, in the language of pseudodierential operators (see, e.g., [8, 10]). Now we want to tackle the following problem: when h evolves according to the k-th KP

ow, how do the currents H j evolve? The answer is given by ( )

2

Proposition 2.2 The KP currents H j evolve in such a way that ( )





@ + H k (H )  H : @tk ( )

+

(2.4)

+

Explicitly, one has

@H j + H k H j = H k @tk ( )

( )

( )

j

( + )

+

k X l=1

j X j (k l) Hl H + Hlk H (j l): l=1

(2.5)

It is important to observe that we can forget about the fact that the currents H i are constructed from h; then equations (2.5) de ne a hierarchy of vector elds in the space H of P i i i sequences fH gi of Laurent series of the form H = z + l Hliz l (i.e., the space of N  N matrices). ( )

( )

( )

0

1

De nition 2.3 The hierarchy de ned by equations (2.5) is called the Central System (CS). It can be shown that the CS ows commute. Another important point is that the CS is a family of dynamical systems in H (i.e., systems of ordinary dierential equations), since the currents H i are no more supposed to depend on the space variable x. ( )

Remark 2.4 The Central System admits some interesting reductions, described in [5]. For

instance, one can obtain the fractional KdV hierarchies [9] and the stationary reductions of the Gelfand-Dickey equations [10] (see the paper by G. Falqui, F. Magri, and G. Tondo in these proceedings). In particular, the KP hierarchy can be seen as the projection of CS on the space of the solutions of the rst CS ow. We will show now that the solutions of CS can be obtained by the solutions of another system (the Sato System ), which is easily linearizable. It is de ned in the same phase-space of CS, but we will use dierent notations for the point, since the twoPsystems are dierent. Hence we consider sequences fW i gi of Laurent series W i = zi + l Wliz l , we introduce the subspace W := hW i ii , and we de ne the Sato System as   @ + zk (W )  W : (2.6) @tk Explicitly, it reads ( )

+

( )

( )

0

1

0

+

@W j + zk W j = W k @tk ( )

( )

j

( + )

+

+

or

@Wmj + W j k @tk

m

+

Wmk j = +

3

k X l=1

k X l=1

Wlj W k l ;

(2.7)

Wlj Wmk l:

(2.8)

(

)

Takasaki has shown ([23], p.29) that these equations represent linear ows in the big cell Gr of the Sato Grassmannian in suitable coordinates. He also showed [22] the connection with the pseudodierential operators formulation of the KP hierarchy. The key point is that P W is the dressing operator, i.e., W exp i tizi is the Baker-Akhiezer function, see, e.g., [8, 17]. The relation with the Central System (2.5) is given by Proposition 2.5 Let  : H ! H be the map de ned as 0

(0)

(0)



W

(i)

1

(

7! H i = ( )

i X l=0

)

Wi l W l =W 0

( )

(0)

:

(2.9)

Then  sends solutions of the Sato System into solutions of the Central System satisfying the constraint H (0) = 1. This result can be cast in the context of Darboux coverings, a concept introduced in [15] to study Darboux transformations (especially for the KP hierarchy). In conclusion, we have reduced the study of the KP equations to the one of the Riccati equations (2.8). Once a solution of (2.8) is known, the corresponding solution of the KP equations (2.3) is simply given by h(x; t) = H (1) (x + t1 ; t2; : : : ), where

H = (W + W W )=W = z + @t@ log W ; (1)

(1)

0 1

(0)

(0)

(2.10)

(0)

1

the last equality being nothing but equation (2.7) for j = 0 and k = 1.

3 Linearization of the Sato System Thanks to the results of the previous section, we can focus on the Sato System. Equations (2.8) can be written in matrix form as @ W + W
T k k
W = W W ; (3.1) k @tk where W = [Wmj ]j ;m and 0

2



6 6 6 6 6 6 = 666 6 6 6 6 4

1

0 1 0


0 0 1 0


... ... ... ... ... ... ... ...

3

2

7 7 7 7 7 7 7 7; 7 7 7 7 7 5

6 6 6 6 6 6 6 6 6 6 6 6 6 4

k=

4

3

0


1 0


7 7 7 ... 1 0





77



1 0 ...

7 7 7 7 7 7 7 7 5

(3.2)

Equation (3.1) is a matrix Riccati equation, which is known [19] to be (formally) linearized by putting W = V
U . Indeed, (3.1) is equivalent to the constant coecients linear system 8 @ U = T k U > > < kV @tk (3.3) > @ V > k : @t =  V 1

k

The second equation is immediately solved, for all k, by X X V (t) = exp( ti i)C = pi(t)i
C ; i1

(3.4)

i0

where t = (t ; t ; : : : ), thePmatrix C is P constant, and the Schur polynomials fpi(t)gi2Z are i de ned, as usual, by exp( i ti z ) = i2Z pi(t)zi (clearly, pi(t) = 0 for i < 0). Then we are left with the following equation for U : X @ U = T k U
pi(t)i
C : (3.5) k @tk i 1

2

1

0

After putting

X

U = exp(

i1

b + D ); ti T i)(UC

(3.6)

where D is another constant matrix, equation (3.5) transforms into @ Ub = ; (3.7) k @tk P P P P where k = exp( i ti T i)
k
exp( i tii) = ( i Pp~iT i)
k
Pi pii, and p~i(t) = pi( t) (or, equivalently, fp~i(t)gi2Z are de ned by exp( i ti zi ) = i2Z p~i (t)zi ). Taking into account the form of k , it is quite easy to show that 1

1

0

0

1

2

6 6 6 6 6

k = 66 6 6 6 4

3

p~

1

p~

2

... ...

k

p~

k

p~

2

3

... ...

k k




p~ 0


77 0

7




p~ 0


777 1




...


...

X

7 exp(

...


7 7 7 ...


5

i1

ti i):

(3.8)

Then, using the fact that @ p~k = p~k j , one immediately proves @tj Proposition 3.1 The solution of equation (3.7) satisfying the initial condition Ub(0) = 0 is X Ub(t) = He
exp( tii); (3.9) i1

where He is the Hankel matrix: Heij = p~i+j 1, for i; j  1.

5

We observe that, although (3.9) is a product of in nitePmatrices, every entry of Ub(t) can be P explicitly computed, thanks to the special form of exp( i tii) = i pi (t)i. Of course, the general solution of equation (3.7) is found by adding a constant matrix to (3.9), but this simply changes the constant matrix D into the expression 1

X

U (t) = exp(

i1

X

ti T i)(He
exp(

i1

tii)C + D):

0

(3.10)

This matrix cannot be explicitly computed (and inverted) unless C and D (i.e., the initial conditions) are carefully chosen. In the next section we will show that this can be done if W (t ) (where t = (t ; t ; : : : ) is the vector of \initial times") has only a nite number of nonzero entries. 0

0

10

20

4 The nite-dimensional case In this section we will write explicitly the general solution of the Sato System in the case where the initial condition W (t ) satis es 0

Wij (t ) = 0 0

for all i > n, j  m.

(4.1)

In Section 5 we will show that in this way we obtain the solutions of the KP hierarchy associated with polynomial  -functions. First of all we observe that (4.1) implies

@Wij (t ) = 0 for all i > n, j  m; (4.2) @tk this means that the ows of the Sato System are tangent to the subspace W m;n of the matrices W such that Wij = 0 for i > n, j  m, and therefore restrict to W m;n . Hence equations (3.1) reduce to Riccati equations for nite matrices, @ Wm;n + W
T 
k ( )k
W = W ( ) W ; (4.3) m;n n;n m;m m;n m;n k n;m m;n @tk where Mm;n denotes the m  n matrix obtained from the in nite matrix M by taking its m  n upper corner. En passant , we remark that from the matrix form (3.1) of the Sato System it also follows that W 2 W m;n =) @@tW = 0 for all k  m + n, (4.4) k so that the solution W (t) of the restriction to W m;n of the Sato System depends only on times (t ; t ;


; tm n ). 0

1

2

+

1

6

For simplicity of notations, let us x m and n, and let us rede ne the matrices appearing in equation (4.3) as

Wm;n = W; m;m = A;

T

n;n = B;

( k )n;m = Gk :

(4.5)

Then equation (4.3) takes the form @W + WB k Ak W = WG W (4.6) k @tk and can still be linearized (as we did in the previous section for the in nite case) by setting W = V U , where V is an m  n matrix and U is an invertible n  n matrix, satisfying 1

8 > >
@V > k : @tk = A V It is not dicult to show that the general solution of this system is obtained by truncating the matrices U (t) and V (t) of Section 3, i.e., that m X1

V (t) = (V (t))m;n = exp(

i=1

tiAi)C

=

m X1 i=0

pi(t)Ai C

(4.8)

m X1 j e U (t) = (U (t))n;n = exp( tj B )(D + H exp( tiAi )C ); j =1 i=1 n 1 X

(4.9)

where C and D are constant (respectively m  n and n  n) matrices and He = Hen;m (i.e., He is the n  m matrix whose entries are Heij = p~i j ). A simpler form of U (t) is obtained by means of +

1

Lemma 4.1 Let H be the n  m matrix de ned by Hij = pi

j

+

n 1 X

m X1 j e exp( tj B )H exp( tiAi ) = j =1 i=1

1

; then

H:

Proof. Equation (4.10) is equivalent to He exp(Pmi ti Ai) = exp( 1

=1

He

m X1 i=0

piAi =

n 1 X j =0

(4.10) Pn

j =1 tj B 1

j )H ,

p~j B j H:

or (4.11)

If we compute the (i; jP) entry of both members of this equation, we P see that it is equivPn m m i p~ p alent to the relations p ~ p = p ~ p , that is, i l j l i r j r l j i l = l r l i Pi Pj i p~r pj i r . But
this follows from l p~l pj i l = 0, which is a consequence of P r i n k
P h k p~k z h ph z = 1. 1

1 = 0

=0

+

+

1 =0 + 1 =0

1

1

0

7

1

+

+

1

1+

=

+

1

Combining equations (4.9) and (4.10), we obtain the following expression for U (t):

U (t) = exp(

n 1 X j =1

tj

B j )D

HC =

n 1 X j =0

pj B j D HC:

 (4.12)

At this point, in order to determine the corresponding solution W (t) of the Sato System (restricted to W m;n ), we should invert U (t) and compute m X1

W (t) = V (t)U (t) = ( 1

i=0

n 1 X

pi(t)Ai)C (

j =0

pj B j D HC ) :

(4.13)

1

In the next section we will present a short cut to compute W (t), based on the introduction of the  -function.

5 The  -function and rational solutions of KP The aim of this section is to show a quicker way to nd the solutions of the Sato System corresponding to the nite-dimensional case discussed in Section 4, and to prove that such solutions lead to rational solutions of the KP hierarchy. The main object is the function  = det U , whose explicit form is

 (t)

=

D21 P 1 p =0

D11

+ p1 D11

n j

m pC j =1 j j 1 Pm j =1 pj +1Cj 1

j Dn j;1

m p j =1 j +n

P






.. .

.. .

.. .

P

m pC D1n j =1 j jn Pm D2n + p1 D1n j =1 pj +1 Cjn






P

j

1C 1






P

n 1p D j =0 j n j;n

m p j =1 j +n

P

1C

jn

:

(5.1)

This expression appears (in a slightly disguised form) in [12], p.338, where it is shown to be the most general polynomial solution of (the Hirota form) of the KP hierarchy. It is clear that one can obtain in particular the  -functions associated with Young diagrams, that is,  = det [pf i j ]i;j ;:::;n, where f  f 


 fn > 0. i

+

1

=1

2

Remark 5.1 The function  is a Wronskian, where derivatives are taken with respect to t ; 1

more precisely, we have that

 (t) = Wr

"n 1 X

j =0

pj Dn

j;n

m X j =1

pj

+

n

1

Cjn; : : : ;

8

n 1 X j =0

pj Dn

j;1

m X j =1

#

pj

n

+

1

Cj : 1

(5.2)

This Wronskian property of  = det U could have been directly shown by multiplying the equation @U = BU G V (5.3) @t by the vectors e = [0; 1; 0; : : : ; 0], e = [0; 0; 1; : : : ; 0], : : : , en = [0; 0; : : : ; 0; 1]. 1

1

2

3

Remark 5.2 In [23], p.29, the evolution equations for the  -function were written. They

can be easily obtained in our formalism noticing that   @ log  = @ log det U = tr @U U = tr B k G W
= tr (G W ) ; k k @tk @tk @tk so that 1

k X

@ log  = @tk

l=1

Wlk l:

(5.4)

(5.5)

Now we want to show that  allows us to determine the whole matrix W (t), i.e., their rows W k . We begin by proving that the link between  and W is expressed by the well-known formula (see, e.g., [8]) ( )

(0)

W (t) =  (t  ([tz) ]) ; 1

(0)

(5.6)

where, as usual, [z ] = (z ; z ; z ; : : : ). First we must recall from [27] the following two lemmas. 1

1 2

1

2

1 3

3

Lemma 5.3 The Schur polynomials satisfy the relations pk (t [z ]) = pk (t) z pk (t): 1

1

(5.7)

1

Lemma 5.4 If v ; v ; : : : ; vj are row vectors in C j and  2 C , then 1

2

+1

+1

2

3

6 vj 6 6 det 66 6 4

vj

+1

...

v

1

v

2

7 7 7 7 7 7 5

2

6 6 6 6 6 = det 66 6 6 6 4

3

1 vj

 .. .

+1

vj .. .

j v

1

9

7 7 7 7 7 7: 7 7 7 7 5

(5.8)

Corollary 5.5 The following formula holds,  (t [z ]) = 1

1 z 1 z 2 . .. z n

m p C j =1 j 1 j 1 Pm D11 j =1 pj Cj 1 Pm D21 + p1 D11 j =1 pj +1Cj 1






P









.. .

n 1p D j =0 j n j;1

m p j =1 j +n

P

P

.. .

j

1C 1






; .. . Pn 1 Pm j =0 pj Dn j;n j =1 pj +n 1Cjn

m p C j =1 j 1 jn Pm D1n j =1 pj Cjn Pm D2n + p1 D1n j =1 pj +1Cjn P

(5.9)

so that the coecient of z l of  (t [z 1 ]), for l = 0; : : : ; n, is given by

(

1)l

P 1 =0

n j

m p C j =1 j 1 j 1 Pm j =1 pj Cj 1

D11

.. .

.. .

(l + 1)-st .. . j;1

P






.. .

pj Dn

m p C j =1 j 1 jn Pm D1n j =1 pj Cjn






P

row .. .

m p j =1 j +n

P

j

1C 1






missing .. . n 1p D j =0 j n j;n

m p j =1 j +n

P

P

1C

Proof. Just combine equation (5.1) with Lemmas 5.3 and 5.4.

jn

:

(5.10)



In order to prove relation (5.6), we have to compute Wl (t) = e W (t)el , where W (t) is given by (4.13) and ei (resp. el ) is the i-th row vector (resp. l-th column vector) of the standard basis. One has 0

1

2

Wl = e Wel = e 0

1

1

m X1 i=0

CU el =  1

10

6 6 6 1 [1; p1; : : : ; pm 1 ]C 66 6 4

Uy 1

...

3

l 7 7

Unly

7 7; 7 7 5

(5.11)

where U y = det(U )U 0

 Wl

=

m

X

i=1

pi

n

X 1

1

is the matrix of algebraic complements. Then

Cik (

1)k+l 

k=1 Pm D11 j =1 pj Cj 1 Pm + p1 D11 j =1 pj +1 Cj 1

D21 P 1 =0 p

n j









.. .

l-th

.. .

row and .. .

k -th

.. .

column .. .

missing .. .

;

(5.12)

Pn Pm


j pj Dn j;n j pj n Cjn which coincides with (5.10). Hence, we have shown that once  (t) is computed via equation (5.1), W can be found using the relation (5.6). Then, all the rows of the matrix W (t) can be recursively computed writing the rst ow of the Sato System, which reads @W j + zW j = W j + W j W ; (5.13) @t and gives W j in terms of the preceding W l . Notice, however, that the corresponding solution of the KP equations (2.3) is simply given by h(x; t) = z + @t@ log W (x + t ; t ; : : : ); (5.14) as explained at the end of Section 2. P

j Dn j;1

m p j =1 j +n

m pC D1n j =1 j jn Pm D2n + p1 D1n j =1 pj +1Cjn P

1 =0

j

1C 1

=1

+

1

(0)

( )

( )

( +1)

( +1)

(0)

1

1

( )

(0)

1

1

2

6 An example In this section we will explicitly nd the solution of the Sato System (2.8) corresponding to the initial conditions W (t ) = W (t ) = 2; Wij (t ) = 0 otherwise; where t = (0; 1; 0; : : : ): (6.1) We already know that we are allowed to consider the restriction of the Sato System to W ; , and that only the times (t ; : : : ; t ) are involved. More precisely, we have to solve equations (4.6) in the case m = 3, n = 2, with initial condition 0 2

0

2 2

0

0

0

32

1

4

3

2 6 6 6 6 6 6 4

0

2

W (t ) = 0 0 0

0

11

2

7 7 7 7: 7 7 5

(6.2)

We also know that the general solution of the system (4.6) is given by W (t) = V (t)U (t) , with (see (4.12) and (5.6)) 1

2 6 6 6 6 6 6 4

3

1 p p 1

2

7 7 7 7 C; 7 7 5

2

V (t) = 0 1 p

1

0 0 1

1 0

6

U (t) = 64

p 1

3

2

7 7D 5

6 6 4

3

p p p 1

2

3

p p p

1

2

3

7 7 C; 5

(6.3)

4

where C (resp. D) is a constant 3  2 (resp. 2  2) matrix. These matrices are determined from the initial value of U and V , that we can choose as V (t ) = W (t ), U (t ) = I . This implies 0

C and therefore

2

V (t)

6 6 6 = 66 6 4

0

0 0 0 0 0

2

7 7 7 7; 7 7 5

2

6

D = 64

3

2p

2

0

2p

0

2

1

4

p p ) = 2t + t 1

3

2 2

4

1 0 0 0

7 7; 5

pp

1 2

p

p

2 1

p

p

1

2

1

1

tt

p 2p

(1)

1

(2)

2

3 7 7 7 7; 7 7 5

(6.6)

t . Hence we have 1

1 2 2 1

1

12

(6.5)

4

W = 1 + 2 (p p z pz ) W = z + 2 (p z pz ) W = z + 2 (p z z ) 1

7 7; 5

1 4 4 1

2 1 2

(0)

3 3

1

6 6 6 6 6 6 4

(6.4)

1 2p

6

U (t) = 64

2

W (t) = 2

3

2

7 7 7 7; 7 7 5

so that

where  = det U = 2(p

0

3

2

6 6 6 = 66 6 4

0

1

1

2

1

1

2

2

2

(6.7)

and, clearly, W compute

k

( )

= zk for all k  3. As an alternative, starting form  we can directly

W =  (t  ([tz) ]) W = @W @t + zW W = @W @t + zW 1

(0)

(1)

(0)

(0)

WW

(0)

(1)

WW

(0)

1

(2)

(1)

1

0 1 1 1

(6.8)

In any case, the corresponding solution of the KP hierarchy is given by (5.14).

Acknowledgments We would like to thank Paolo Casati for useful discussions and comments, and the anonymous referee for helpful remarks.

References [1] M.J. Ablowitz, J. Satsuma, Soliton and rational solutions of nonlinear evolution equations. J. Math. Phys. 19 (1978), 2180{2186. [2] H. Airault, H.P. McKean, J. Moser, Rational and elliptic solutions of the Korteweg-de Vries equation and a related many-body problem. Commun. Pure Appl. Math. 30 (1977), 95{148. [3] The Bispectral Problem (Montreal, PQ, 1997), J. Harnad and A. Kasman eds., 1998, Amer. Math. Soc., Providence, RI. [4] F. Calogero, Motion of poles and zeroes of special solutions of nonlinear and linear partial dierential equations, and related \solvable" many{body problems. Nuovo Cimento 43B (1978), 177{241. [5] P. Casati, G. Falqui, F. Magri, M. Pedroni, A note on Fractional KdV Hierarchies. J. Math. Phys. 38 (1997), 4606{4628. [6] I. V. Cherednik, Dierential equations for the Baker-Akhiezer functions of Algebraic curves. Funct. Anal. Appl. 12 (1978), 45{54 (Russian), 195{203 (English). [7] D.V. Chudnovsky, G.V. Chudnovsky, Pole expansions of nonlinear partial dierential equations. Nuovo Cimento 40B (1977), 339{353. 13

[8] E. Date, M. Jimbo, M. Kashiwara, T. Miwa, Transformation Groups for Soliton Equations. Proceedings of R.I.M.S. Symposium on Nonlinear Integrable Systems{Classical Theory and Quantum Theory (M. Jimbo, T. Miwa, eds.), World Scienti c, Singapore, 1983, pp. 39{119. [9] M.F. de Groot, T.J. Hollowood, J.L. Miramontes, Generalized Drinfel'd-Sokolov Hierarchies. Commun. Math. Phys. 145 (1992), 57{84. [10] L. A. Dickey, Soliton Equations and Hamiltonian Systems. Adv. Series in Math. Phys. Vol. 12, World Scienti c, Singapore, 1991. [11] G. Falqui, F. Magri, M. Pedroni, Bihamiltonian Geometry, Darboux Coverings, and Linearization of the KP Hierarchy. Commun. Math. Phys. 197 (1998), 303{324 [12] V.G. Kac, In nite dimensional Lie algebras , third edition. Cambridge University Press, 1990. [13] I.M. Krichever, On rational solutions of the Kadomstev-Petviashvili equation and integrable systems of N particles on the line. Funct. Anal. Appl. 12 (1978), 76{78 (Russian), 59{61 (English). [14] M.D. Kruskal, The Korteweg-de Vries equation and related evolution equations. In: Nonlinear Wave Motion (A.C. Newell, ed.), Lect. Appl. Math. vol. 15, AMS, Providence, 1974, pp. 61{83. [15] F. Magri, M. Pedroni, J.P. Zubelli, On the Geometry of Darboux Transformations for the KP Hierarchy and its Connection with the Discrete KP Hierarchy. Commun. Math. Phys. 188 (1997), 305{325. [16] V.B. Matveev, Some comments on the rational solutions of the Zakharov-Schabat equations. Lett. Math. Phys. 3 (1979), 503{512. [17] M. Mulase, Algebraic Theory of the KP Equations. Perspectives in Mathematical Physics (R. Penner and S{T. Yau, eds.), International Press, Boston, 1994, pp. 151{217. [18] Y. Ohta, J. Satsuma, D. Takahashi, T. Tokihiro, An Elementary Introduction to Sato Theory. In: Recent Developments in Soliton Theory (J. Satsuma et al. eds.), Progr. Theoret. Phys. Suppl. 94 (1988), 210{241. [19] W.T. Reid, Riccati Dierential Equations. Academic Press, New York, 1972. [20] M. Sato, Y. Sato, Soliton equations as dynamical systems on in nite{dimensional Grassmann manifold. In: Nonlinear PDEs in Applied Sciences (US-Japan Seminar, Tokyo), P. Lax and H. Fujita eds., North-Holland, Amsterdam, 1982, pp. 259{271. 14

[21] T. Shiota, Calogero-Moser hierarchy and KP hierarchy. J. Math. Phys. 35 (1994), 5844{ 5849. [22] K. Takasaki, Integrable System as deformations of D-modules. In: Theta Functions { Bowdoin 1987 (L. Ehrenpreis and R.C. Gunning, eds.), Proc. Symp. Pure Math. vol. 49 (1989), AMS, pp. 143{168. [23] K. Takasaki, Geometry of Universal Grassmann Manifold from Algebraic point of view. Rev. Math. Phys. 1 (1989), 1{46. [24] G. Wilson, On two Constructions of Conservation Laws for Lax equations. Quart. J. Math. Oxford 32 (1981), 491{512. [25] G. Wilson, Bispectral commutative ordinary dierential operators. J. Reine Angew. Math. 442 (1993), 177{204. [26] G. Wilson, Collisions of Calogero-Moser particles and an adelic Grassmannian. Invent. Math. 133 (1998), 1{41. [27] J.P. Zubelli, On the polynomial  -functions for the KP hierarchy and the bispectral property. Lett. Math. Phys. 24 (1992), 41{48.

15