JHEP 02 (2007) 094hep-th/0610006

October, 2006

Notes on Exact Multi-Soliton Solutions of

Noncommutative Integrable Hierarchies

(Extended Version)

Masashi Hamanaka1

Graduate School of Mathematics, Nagoya University,Chikusa-ku, Nagoya, 464-8602, JAPAN

Mathematical Institute, University of Oxford,24-29, St Giles’, Oxford, OX1 3LB, UK

Abstract

We study exact multi-soliton solutions of integrable hierarchies on noncommutative space-times which are represented in terms of quasi-determinants of Wronski matrices by Etingof,Gelfand and Retakh. We analyze the asymptotic behavior of the multi-soliton solutionsand found that the asymptotic configurations in soliton scattering process can be all thesame as commutative ones, that is, the configuration of N -soliton solution has N iso-lated localized energy densities and the each solitary wave-packet preserves its shape andvelocity in the scattering process. The phase shifts are also the same as commutativeones. Furthermore noncommutative toroidal Gelfand-Dickey hierarchy is introduced andthe exact multi-soliton solutions are given.

In this extended version, we add proofs of some results by Etingof, Gelfand and Retakh,so that this paper becomes more self-contained. Discussion on conservation laws are alsoreviewed in an addtional section.

1The author visits Oxford from 16 August, 2005 to 15 December, 2006.E-mail: hamanaka@math.nagoya-u.ac.jp

1 Introduction

Extension of integrable systems and soliton theories to non-commutative (NC) space-

times 2 have been studied by many authors for the last couple of years and various kind

of integrable-like properties have been revealed [1, 2]. This is partially motivated by

recent developments of NC gauge theories on D-branes. In the NC gauge theories, NC

extension corresponds to introduction of background magnetic fields and NC solitons are,

in some situations, just lower-dimensional D-branes themselves. Hence exact analysis of

NC solitons just leads to that of D-branes and various applications to D-brane dynamics

have been successful [3]. In this sense, NC solitons plays important roles in NC gauge

theories.

Most of NC integrable equations such as NC KdV equations apparently belong not

to gauge theories but to scalar theories. However now, it is proved that they can be

derived from NC anti-self-dual (ASD) Yang-Mills equations by reduction [4], which is

first conjectured explicitly by the author and K. Toda [5]. (Original commutative one

is proposed by R. Ward [6] and hence this conjecture is sometimes called NC Ward’s

conjecture. For more about commutative one, see e.g. [7]-[10].) Therefore analysis of

exact soliton solutions of NC integrable equations could be applied to the corresponding

physical situations in the framework of N=2 string theory [11, 12, 13].

Furthermore, some soliton equations describe real phenomena such as shallow water

waves in fluid dynamics, optics and so on. If noncommutativity in space-time affects

soliton dynamics, then we can check whether our universe is noncommutative or not by

comparing experimental results and estimate the strength or the upper bound of the

noncommutativity

Hence, construction and analysis of exact multi-soliton solutions are worth studying

from various viewpoints of integrable systems, string theory, and perhaps detection of

noncommutativity in our universe.

Exact multi-soliton solutions of noncommutative KP hierarchy are constructed by

Etingof, Gelfand and Retakh in 1997 [14], where quasi-determinants play crucial roles.

(For other applications of quasi-determinants to noncommutative integrable systems, see

e.g. [15]-[21].) However, their discussion is general and explicit analysis of the behavior

of their soliton solutions has not yet been done. Paniak also constructs multi-soliton

solutions of NC KP and KdV equations (not hierarchies) and studies the scattering process

[23].3 However, the discussion about the soliton dynamics is mainly focused on two-soliton

2In the present paper, the word “NC” always refers to generalization to noncommutative spaces, notto non-abelian and so on.

3Dimakis and Muller-Hoissen present perturbative corrections with respect to a noncommutative pa-

1

scatterings.

In this paper, we study exact multi-soliton solutions of NC integrable hierarchies in

terms of quasi-determinants of Wronski matrices, which is developed by Etingof, Gelfand

and Retakh. We analyze the asymptotic behavior of the multi-soliton solutions and found

that the asymptotic configurations can be real-valued though NC fields take complex

values in general. The behavior in soliton scatterings is all the same as commutative

ones, that is, the N -soliton solution has N isolated localized energy densities and the

each wave-packet preserve its shape and velocity in the scattering process. The phase

shift is also the same as commutative one.

This paper is organized as follows. In section 2, we make a brief introduction to NC

field theory in star-product formalism. In section 3 and 4, we review definition and some

properties of quasi-determinants and their applications to construction of multi-soliton

solutions of NC integrable hierarchy in star-product formalism. In the end of section 4,

we introduce NC toroidal Gelfand-Dickey (GD) hierarchy and give exact multi-soliton

solutions which are new. In section 5, we discuss the asymptotic behavior of them in

detail. In section 6, we prove existence of infinite conserved densities of NC KP hierarchy

by presenting explicit representations of them. Section 7 is devoted to conclusion and

discussion.

2 NC Field Theory in the star-product formalism

NC spaces are defined by the noncommutativity of the coordinates:

[xi, xj] = iθij, (2.1)

where the constant θij is called the NC parameter. If the coordinates are real, NC pa-

rameters should be real. Because the rank of the NC parameter is even, dimension of NC

space-times must be more than two. Hence in this paper, we deal not with integrable

systems in (0 + 1)-dimension such as the Painleve equation, but with ones in (1 + 1)

or (2 + 1)-dimension such as the KdV and KP equations. In (1 + 1)-dimension, we can

take only space-time noncommutativity as [t, x] = iθ. In (2 + 1)-dimension, there are

essentially two kind of choices of noncommutativity, that is, space-space noncommuta-

tivity: [x, y] = iθ and space-time noncommutativity: [t, x] = iθ or [t, y] = iθ, where the

coordinates (x, y) and t correspond to space and time coordinates, respectively.

NC field theories are given by the replacement of ordinary products in the commu-

tative field theories with the star-products and realized as deformed theories from the

rameter in 2-soliton scatterings of the NC KdV equation [24] before the Paniak’s work.

2

commutative ones. The star-product is defined for ordinary fields on flat spaces, explic-

itly by

f ? g(x) := exp(

i

2θij∂

(x′)i ∂

(x′′)j

)f(x′)g(x′′)

∣∣∣x′=x′′=x

= f(x)g(x) +i

2θij∂if(x)∂jg(x) +O(θ2), (2.2)

where ∂(x′)i := ∂/∂x′i and so on. This explicit representation is known as the Moyal product

[25]. The ordering of fields in nonlinear terms are determined so that some structures such

as gauge symmetries and Lax representations should be preserved.

The star-product has associativity: f ? (g ?h) = (f ?g)?h, and reduces to the ordinary

product in the commutative limit: θij → 0. The modification of the product makes the

ordinary spatial coordinate “noncommutative,” that is, [xi, xj]? := xi ? xj − xj ? xi = iθij.

We note that the fields themselves take c-number values as usual and the differenti-

ation and the integration for them are well-defined as usual. A nontrivial point is that

NC field equations contain infinite number of derivatives in general. Hence the integra-

bility of the equations are not so trivial as commutative cases, especially for space-time

noncommutativity.

3 Brief Review of Quasi-determinants

In this section, we make a brief introduction of quasi-determinants introduced by Gelfand

and Retakh [26, 27] and present a few properties of them which play important roles in

the following sections. The detailed discussion is seen in e.g. [28, 29]. Relation between

quasi-determinants and NC symmetric functions is seen in e.g. [30].

Quasi-determinants are not just a generalization of usual commutative determinants

but rather related to inverse matrices. From now on, we suppose existence of all the

inverses.

Let A = (aij) be a N × N matrix and B = (bij) be the inverse matrix of A, that is,

A?B = B?A = 1. Here all products of matrix elements are supposed to be star-products,

though the present discussion hold for more general situation where the matrix elements

belong to a noncommutative ring.

Quasi-determinants of A are defined formally as the inverse of the elements of B = A−1:

|A|ij := b−1ji . (3.1)

In the commutative limit, this is reduced to

|A|ij −→ (−1)i+j det A

det Aij, (3.2)

3

where Aij is the matrix obtained from A deleting the i-th row and the j-th column.

We can write down more explicit form of quasi-determinants. In order to see it, let us

recall the following formula for a block-decomposed square matrix:(

A BC D

)−1

=

((A−B ? D−1 ? C)−1 −A−1 ? B ? (D − C ? A−1 ? B)−1

−(D − C ? A−1 ? B)−1 ? C ? A−1 (D − C ? A−1 ? B)−1

),

where A and D are square matrices. We note that any matrix can be decomposed as

a 2 × 2 matrix by block decomposition where one of the diagonal parts is 1 × 1. Then

the above formula can be applied to the decomposed 2× 2 matrix and an element of the

inverse matrix is obtained. Hence quasi-determinants can be also given iteratively by:

|A|ij = aij −∑

i′(6=i),j′(6=j)

aii′ ? ((Aij)−1)i′j′ ? aj′j

= aij −∑

i′(6=i),j′(6=j)

aii′ ? (|Aij|j′i′)−1 ? aj′j. (3.3)

It is sometimes convenient to represent the quasi-determinant as follows:

|A|ij =

a11 · · · a1j · · · a1n...

......

ai1 aij ain

......

...an1 · · · anj · · · ann

. (3.4)

Examples of quasi-determinants are, for a 1× 1 matrix A = a

|A| = a,

and for a 2× 2 matrix A = (aij)

|A|11 =a11 a12

a21 a22= a11 − a12 ? a−1

22 ? a21, |A|12 =a11 a12

a21 a22= a12 − a11 ? a−1

21 ? a22,

|A|21 =a11 a12

a21 a22= a21 − a22 ? a−1

12 ? a11, |A|22 =a11 a12

a21 a22= a22 − a21 ? a−1

11 ? a12,

and for a 3× 3 matrix A = (aij)

|A|11 =

a11 a12 a13

a21 a22 a23

a31 a32 a33

= a11 − (a12, a13) ?

(a22 a23

a32 a33

)−1

?

(a21

a31

)

= a11 − a12 ?a22 a23

a32 a33

−1

? a21 − a12 ?a22 a23

a32 a33

−1

? a31

− a13 ?a22 a23

a32 a33

−1

? a21 − a13 ?a22 a23

a32 a33

−1

? a31,

4

and so on.

Quasi-determinants have the following properties:

Proposition 3.1 [26] Let A = (aij) be a square matrix of order n.

(i) Permutation of Rows and Columns.

The quasi-determinant |A|ij does not depend on permutations of rows and columns in

the matrix A that do not involve the i-th row and j-th column.

(ii) The multiplication of rows and columns.

Let the matrix M = (mij) be obtained from the matrix A by multiplying the i-th row

by f(x) from the left, that is, mij = f ? aij and mkj = akj for k 6= i. Then

|M |kj =

{f ? |A|ij for k = i|A|kj for k 6= i

(3.5)

Let the matrix N = (nij) be obtained from the matrix A by multiplying the j-th

column by f(x) from the right, that is, nij = aij ? f and nil = ail for l 6= j. Then

|N |il =

{|A|ij ? f for l = j|A|il for l 6= j

(3.6)

(iii) The addition of rows and columns.

Let the matrix M = (mij) be obtained from the matrix A by replacing the k-th row

of A with the sum of the k-th row and l-th row, that is, mkj = akj + ajl and mij = aij for

k 6= i. Then

|A|ij = |M |ij, for i 6= k. (3.7)

Let the matrix N = (nij) be obtained from the matrix A by replacing the k-th column

of A with the sum of the k-th column and l-th column, that is, nik = aik +ail and nij = aij

for k 6= j. Then

|A|ij = |N |ij, for j 6= k. (3.8)

Proposition 3.2 [26] If the quasi-determinant |A|ij is defined, then the following state-

ments are equivalent.

(i) |A|ij = 0.

(ii) the i-th row of the matrix A is a left linear combination of the other rows of A.

(iii) the j-th column of the matrix A is a right linear combination of the other columns

of A.

5

4 Exact Soliton Solutions of NC Integrable Hierar-

chies

In this section, we give exact multi-soliton solutions of several NC integrable hierarchies in

terms of quasi-determinants. In the commutative case, determinants of Wronski matrices

play crucial roles. In the NC case, these determinants are just replaced with the quasi-

determinants. We review foundation of the NC KP hierarchy and the l-reduced hierarchies

(so called NC GD hierarchies or NC lKdV hierarchies), and present the exact multi-soliton

solutions of them developed by Etingof, Gelfand and Retakh [14]. Finally we extend their

discussion to the NC toroidal GD hierarchy.

4.1 Pseudo-differential operators

An N -th order pseudo-differential operator A is represented as follows

A = aN∂Nx + aN−1∂

N−1x + · · ·+ a0 + a−1∂

−1x + a−2∂

−2x + · · · , (4.1)

where ai is a function of x associated with noncommutative associative products (here,

the Moyal products). When the coefficient of the highest order aN equals to 1, we call it

monic. Here we introduce useful symbols:

A≥r := ∂Nx + aN−1∂

N−1x + · · ·+ ar∂

rx, (4.2)

A≤r := A− A≥r+1 = ar∂rx + ar−1∂

r−1x + · · · , (4.3)

resrA := ar. (4.4)

The symbol res−1A is especially called the residue of A.

The action of a differential operator ∂nx on a multiplicity operator f is formally defined

as the following generalized Leibniz rule:

∂nx · f :=

∑

i≥0

(ni

)(∂i

xf)∂n−i, (4.5)

where the binomial coefficient is given by

(ni

):=

n(n− 1) · · · (n− i + 1)

i(i− 1) · · · 1 . (4.6)

We note that the definition of the binomial coefficient (4.6) is applicable to the case for

negative n, which just define the action of negative power of differential operators.

6

The examples are,

∂−1x · f = f∂−1

x − f ′∂−2x + f ′′∂−3

x − · · · ,∂−2

x · f = f∂−2x − 2f ′∂−3

x + 3f ′′∂−4x − · · · ,

∂−3x · f = f∂−3

x − 3f ′∂−4x + 6f ′′∂−5

x − · · · , (4.7)

where ∂−1x in the RHS acts on a function as an integration

∫ x dx.

The composition of pseudo-differential operators is also well-defined and the total set of

pseudo-differential operators forms an operator algebra. For a monic pseudo-differential

operator A, there exist the unique inverse A−1 and the unique m-th root A1/m which

commute with A. (These proofs are all the same as commutative ones.) For more on

pseudo-differential operators and Sato’s theory, see e.g. [31, 32, 33, 34].

4.2 NC KP and GD hierarchies

In order to define the NC KP hierarchy, let us introduce a Lax operator:

L = ∂x + u2∂−1x + u3∂

−2x + u4∂

−3x + · · · , (4.8)

where the coefficients uk (k = 2, 3, . . .) are functions of infinite coordinates ~x := (x1, x2, . . .)

with x1 ≡ x:

uk = uk(x1, x2, . . .). (4.9)

The noncommutativity is introduced into the coordinates (x1, x2, . . .) as Eq. (2.1) here.

The NC KP hierarchy is defined in Sato’s framework as

∂mL = [Bm, L]? , m = 1, 2, . . . , (4.10)

where the action of ∂m := ∂/∂xm on the pseudo-differential operator L should be in-

terpreted to be coefficient-wise, that is, ∂mL := [∂m, L]? or ∂m∂kx = 0. The differential

operator Bm is given by

Bm := (L ? · · · ? L︸ ︷︷ ︸m times

)≥0 =: (Lm)≥0. (4.11)

The KP hierarchy gives rise to a set of infinite differential equations with respect to infinite

kind of fields from the coefficients in Eq. (4.10) for a fixed m. Hence it contains huge

amount of differential (evolution) equations for all m. The LHS of Eq. (4.10) becomes

∂muk which shows a kind of flow in the xm direction.

7

If we put the constraint (Ll)≤−1 = 0 or equivalently Ll = Bl on the NC KP hierarchy

(4.10), we get a reduced NC KP hierarchy which is called the l-reduction of the NC KP

hierarchy, or the NC lKdV hierarchy, or the l-th NC Gelfand-Dickey (GD) hierarchy.

Especially, the 2-reduction of NC KP hierarchy is just the NC KdV hierarchy.

We can easily show

∂uk

∂xnl

= 0, (4.12)

for all n, k because ∂Ll/∂xnl = [Bnl, Ll] = [(Ll)n, Ll] = 0, which implies Eq. (4.12). This

time, the constraint Ll = Bl gives simple relationships which make it possible to represent

infinite kind of fields ul+1, ul+2, ul+3, . . . in terms of (l− 1) kind of fields u2, u3, . . . , ul. (cf.

Appendix A in [36].)

Let us see explicit examples.

• NC KP hierarchy

The coefficients of each powers of (pseudo-)differential operators in the NC KP

hierarchy (4.10) yield a series of infinite NC “evolution equations,” that is, for

m = 1

∂1−kx ) ∂1uk = u′k, k = 2, 3, . . . ⇒ x1 ≡ x, (4.13)

for m = 2

∂−1x ) ∂2u2 = u′′2 + 2u′3,

∂−2x ) ∂2u3 = u′′3 + 2u′4 + 2u2 ? u′2 + 2[u2, u3]?,

∂−3x ) ∂2u4 = u′′4 + 2u′5 + 4u3 ? u′2 − 2u2 ? u′′2 + 2[u2, u4]?,

∂−4x ) ∂2u5 = · · · , (4.14)

and for m = 3

∂−1x ) ∂3u2 = u′′′2 + 3u′′3 + 3u′4 + 3u′2 ? u2 + 3u2 ? u′2,

∂−2x ) ∂3u3 = u′′′3 + 3u′′4 + 3u′5 + 6u2 ? u′3 + 3u′2 ? u3 + 3u3 ? u′2 + 3[u2, u4]?,

∂−3x ) ∂3u4 = u′′′4 + 3u′′5 + 3u′6 + 3u′2 ? u4 + 3u2 ? u′4 + 6u4 ? u′2

−3u2 ? u′′3 − 3u3 ? u′′2 + 6u3 ? u′3 + 3[u2, u5]? + 3[u3, u4]?,

∂−4x ) ∂3u5 = · · · . (4.15)

These just imply the (2+1)-dimensional NC KP equation [23, 31] with 2u2 ≡ u, x2 ≡y, x3 ≡ t and ∂−1

x f(x) =∫ x dx′f(x′):

∂u

∂t=

1

4

∂3u

∂x3+

3

4

∂(u ? u)

∂x+

3

4∂−1

x

∂2u

∂y2− 3

4

[u, ∂−1

x

∂u

∂y

]

?

. (4.16)

8

Important point is that infinite kind of fields u3, u4, u5, . . . are represented in terms

of one kind of field 2u2 ≡ u as is seen in Eq. (4.14). This guarantees the existence of

NC KP hierarchy which implies the existence of reductions of the NC KP hierarchy.

The order of nonlinear terms are determined in this way.

• NC KdV Hierarchy (2-reduction of the NC KP hierarchy)

Taking the constraint L2 = B2 =: ∂2x + u for the NC KP hierarchy, we get the NC

KdV hierarchy. This time, the following NC hierarchy

∂u

∂xm=

[Bm, L2

]?, (4.17)

include neither positive nor negative power of (pseudo-)differential operators for the

same reason as commutative case and gives rise to the m-th KdV equation for each

m. For example, the NC KdV hierarchy (4.17) becomes the (1+1)-dimensional NC

KdV equation [24] for m = 3 with x3 ≡ t

u =1

4u′′′ +

3

4(u′ ? u + u ? u′) , (4.18)

and the (1 + 1)-dimensional 5-th NC KdV equation [37] for m = 5 with x5 ≡ t

u =1

16u′′′′′ +

5

16(u ? u′′′ + u′′′ ? u) +

5

8(u′ ? u′ + u ? u ? u)′. (4.19)

In this way, we can generate infinite set of the l-reduced NC KP hierarchies. Explicit

examples are seen in e.g. [36]. (See also [38, 40].)

4.3 Exact multi-soliton solutions of NC KP hierarchy

Before discussing the exact multi-soliton solutions of the NC KP hierarchy, we present a

theorem on factorization of a differential operator of order n

L = ∂nx + a1∂

n−1x + · · ·+ an−1∂x + an, (4.20)

where the coefficient ai is a function of x, which is essential for construction of N -

soliton solutions of NC integrable hierarchies. Let us introduce the Wronski matrix

W (f1, f2, · · · , fm) as usual:

W (f1, f2, · · · , fm) :=

f1 f2 · · · fm

f ′1 f ′2 · · · f ′m...

.... . .

...

f(m−1)1 f

(m−1)2 · · · f (m−1)

m

, (4.21)

9

where f1, f2, · · · , fm are functions of x and f ′ := ∂f/∂x, f ′′ := ∂2f/∂x2, f (m) := ∂mf/∂xm

and so on. Then the following theorem holds:

Theorem 4.1 [14]

(i) For any non-degenerate set of elements f1, . . . , fn, there exists a unique monic dif-

ferential operator of order n such that L ? fi = 0 for i = 1, · · · , n. It is given by the

formula

L ? f = |W (f1, . . . , fn, f)|n+1,n+1. (4.22)

(ii) Let L be an n-th order monic differential operator and f1, . . . , fn be a set of solutions

of L ? fi = 0, such that for any m < n the set of elements f1, . . . , fm is non-degenerate.

Then L admits a factorization L = (∂x − bn) ? · · · ? (∂x − b1) where

bi = W ′i ? W−1

i , Wi := |W (f1, . . . , fi)|ii. (4.23)

(proof) (i) The operator L can be written as

L = ∂nx + (an, an−1, · · · , a1)

1∂x...

∂n−1x

. (4.24)

Hence the condition L ? (f1, f2, · · · , fn) = 0 becomes

(f(n)1 , f

(n)2 , · · · , f (n)

n ) + (an, an−1, · · · , a1) ?

f1 f2 · · · fn

f ′1 f ′2 · · · f ′n...

... . . ....

f(n−1)1 f

(n−1)2 · · · f (n−1)

n

︸ ︷︷ ︸=W (f1,f2,···,fn)

= 0, (4.25)

and we get

(an, an−1, · · · , a1) = −(f(n)1 , f

(n)2 , · · · , f (n)

n ) ? W (f1, f2, · · · , fn)−1. (4.26)

On the other hand,

|W (f1, · · · , fn, f)|n+1,n+1 = f (n) − (f(n)1 , f

(n)2 , · · · , f (n)

n ) ? W (f1, f2, · · · , fn)−1 ?

ff ′...

f (n−1)

= f (n) + (an, an−1, · · · , a1) ?

ff ′...

f (n−1)

= L ? f Q.E.D.

10

(ii) Let us prove the statement by induction in n. For n = 1, it is obvious. Supose that it

is valid for the differential operator Ln−1 which satisfies Ln−1 ? fi = 0 for i = 1, · · · , n− 1.

Due to (i), this operator exists and is unique. Now let us consider the operator L :=

(∂x − bn) ? Ln−1 where bn := W ′n ? W−1

n . It is obvious that L ? fi = 0 for i = 1, · · · , n− 1.

Furthermore, by (i),

L ? fn = L ? fn = (∂x − bn) ? Ln−1 ? fn = (∂x − bn) ? |W (f1, · · · , fn−1, fn)|n,n

= (∂x − bn) ? Wn = W ′n −W ′

n ? W−1n ? Wn = 0. (4.27)

Therefore by (i), L = L. Q. E. D.

Now we construct multi-soliton solutions of the NC KP hierarchy. Let us introduce

the following functions,

fs(~x) = eξ(~x;αs)? + ase

ξ(~x;βs)? , (4.28)

where

ξ(~x; α) = x1α + x2α2 + x3α

3 + · · · , (4.29)

and αs, βs and as are constants. Star exponential functions are defined by

ef(x)? := 1 +

∞∑

n=1

1

n!f(x) ? · · · ? f(x)︸ ︷︷ ︸

n times

. (4.30)

Theorem 4.2 An N -soliton solution of the NC KP hierarchy (4.10) is given by [14],

L = ΦN ? ∂xΦ−1N , (4.31)

where

ΦN ? f = |W (f1, . . . , fN , f)|N+1,N+1,

=

f1 f2 · · · fN ff ′1 f ′2 · · · f ′N f ′...

.... . .

......

f(N−1)1 f

(N−1)2 · · · f

(N−1)N f (N−1)

f(N)1 f

(N)2 · · · f

(N)N f (N)

. (4.32)

(Proof) The proof is the same as propositon 5.3.7 in [33]. L = ΦN ? ∂xΦ−1N implies

Lm = ΦN ? ∂mx Φ−1

N , or equivalently Lm ? ΦN = ΦN∂mx . Hence

Lm≥0 ? ΦN − ΦN∂m

x = −Lm≤−1 ? ΦN . (4.33)

11

By proposition 3.2, ΦN ? fs = |W (f1, . . . , fN , fs)|N+1,N+1 = 0 for s = 1, · · · , N . Hence

0 = ∂m(ΦN ? fs) = (∂mΦN) ? fs + ΦN ? (∂mfs︸ ︷︷ ︸=∂m

x fs

)

(4.33)= (∂mΦN) ? fs + Lm

≥0 ? ΦN ? fs︸ ︷︷ ︸=0

+ Lm≤−1 ? ΦN ? fs︸ ︷︷ ︸

6=0

= (∂mΦN + Lm≤−1 ? ΦN) ? fs. (4.34)

We note that because s runs from 1 to N and ∂mΦN + Lm≤−1 ? ΦN is less than order n, we

get

∂mΦN + Lm≤−1 ? ΦN = 0. (4.35)

Therefore

∂mL = (∂mΦN) ? ∂xΦ−1N − ΦN ? ∂xΦ

−1N ? ∂mΦN ? Φ−1

N

(4.35)= −Lm

≤−1 ? ΦN ? ∂xΦ−1N − ΦN ? ∂xΦ

−1N ? Lm

≤−1 ? ΦN ? Φ−1N︸ ︷︷ ︸

=1

= [ΦN ? ∂xΦ−1N , Lm

≤−1]? = [Lm≥0, ΦN ? ∂xΦ

−1N ]?. = [Bm, L]? Q.E.D.

In the commutative limit, ΦN ? f is reduced to

ΦN ? f −→ det W (f1, f2, . . . , fN , f)

det W (f1, f2, . . . , fN), (4.36)

which just coincides with commutative one [33]. In this respect, quasi-determinants are

fit to this framework of Wronskian solutions.

By comparing the coefficient of ∂N−1x in L ? ΦN = ΦN∂x and applying Theorem 4.1

(ii) to the n-th order monic differential operator ΦN , we have a more explicit form of the

N -soliton solution as

u ≡ 2u2 = 2∂x

(N∑

s=1

bs

)= 2∂x

(N∑

s=1

W ′s ? W−1

s

). (4.37)

The l-reduction condition (Ll)≤−1 = 0 or Ll = Bl is realized at the level of the soliton

solutions by taking αls = βl

s or equivalently αs = εβs for s = 1, · · · , N , where ε is the l-th

root of unity. The proof is as follows.

First we note that ∂lxfs = αl

sfs because of the condition αls = βl

s. Hence

(Ll≥0 ? ΦN − ΦN∂l

x) ? fs = 0. (4.38)

12

On the other hand, Ll = ΦN ? ∂lxΦN implies

Ll≥0 ? ΦN − ΦN∂l

x = −Ll≤−1 ? ΦN . (4.39)

Because the RHS is less than order N and the LHS is so. Hence due to (4.38), the LHS

is identidcally zero and the RHS is so: Ll≤−1 ? ΦN = 0. The operator ΦN is monic and

invertible, and therefore we get Ll≤−1 = 0 which is the l-reduction condition.

4.4 Exact multi-soliton solutions of NC toroidal GD hierarchy

The present discussion is straightforwardly applicable for NC versions of the matrix KP

hierarchy [41, 32, 33], the toroidal (matrix) GD hierarchy [42, 43, 44, 45, 46, 47] and the

(2-dimensional) Toda lattice hierarchy [48] formulated by pseudo-differential operators,

because on commutative spaces, their exact soliton solutions are described by determi-

nants of (generalized) Wronski matrices.

For example, we can give exact N -soliton solutions of the NC toroidal lKdV hierarchy

(l ≥ 2)4 which is defined as follows.

First, we introduce two kind of infinite variables ~x = (x1, x2, · · ·) and ~y := (y0, yl, y2l, · · ·)with (x, y) ≡ (x1, y0). Noncommutativity is introduced into these coordinates. Next let

us define two kind of Lax operators with respect to x, that is, an l-th order differential

operator P = (L)l≥0 and a 0-th order pseudo-differential operator Q, where the coeffi-

cients depend on the two kind of infinite variables. An differential operator Cml is also

introduced in terms of P and Q as Cml := −(Pm ? Q)≥0. Then we can obtain the NC

toroidal lKdV hierarchy:

∂P

∂xm

= [Bm, P ]? ,∂Q

∂xm

= [Bm, Q− ∂y]? , (4.40)

∂P

∂yml

= [Pm∂y + Cml, P ]? ,∂Q

∂yml

= [Pm∂y + Cml, Q− ∂y]? . (4.41)

For l = 2, this includes the NC Calogero-Bogoyavlenskii-Schiff equation [37].

The N -soliton solution is given by

P = ΦN ? ∂lxΦ

−1N , Q = (∂yΦN) ? Φ−1

N , (4.42)

4Toroidal lKdV hierarchy is one of generalizations of lKdV hierarchy and first studied by Bogoy-avlenskii [42] for l = 2 and developed by Billig, Iohara, Saito, Wakimoto, Ikeda and Takasaki wherethe symmetry of the solution space is revealed to be described in terms of a toroidal Lie algebra, thatis, a central extension of double loop algebra Gtor = SLtor

l [44, 45, 46]. Hence we call it toroidal lKdVhierarchy or toroidal GD hierarchy in the present paper.

13

where the arguments in ΦN is modified as follows:

fs(~x, ~y) := eξrs (~x,~y;αs)? + ase

ξrs (~x,~y;βs)? , (4.43)

ξr(~x, ~y; α) := ξ(~x; α) + rξ(~y; α)

= x1α + x2α2 + x3α

3 + · · ·+ ry0 + rylαl + ry2lα

2l + · · · , (4.44)

with αls = βl

s, where r is a constant. The proof is the same as the commutative one. (For

the details, see section 5.1 in [46].) A key point of the proof is to show the evolution

equations of ΦN :

∂ΦN

∂xm

= −(ΦN∂mx Φ−1

N )≤−1 ? ΦN = Bm ? ΦN − ΦN∂mx ,

∂ΦN

∂yml

= (Pm ? Q)≤−1 ? ΦN = (Pm∂y + Cml) ? ΦN ,

where the following property of quasideterminant plays crucial roles:

ΦN ? fs = |W (f1, . . . , fN , fs)|N+1,N+1 = 0, for s = 1, · · · , N.

This hierarchy generally gives rise to (2+1)-dimensional integrable equations where space

and time coordinates are (x, y) and some other coordinate, respectively.

5 Asymptotic Behavior of the Exact Soliton Solu-

tions

In this section, we discuss asymptotic behavior of the multi-soliton solutions at spatial

infinity or infinitely past and future. Here we restrict ourselves to NC KdV and KP

hierarchies, however, this observation would be also applicable to other NC hierarchies.

First, we present some special properties of the star exponential functions relevant

to behavior of NC soliton solutions. In this section, we restrict ourselves to a specific

equation on (2 + 1) or (1 + 1)-dimensional space-time and noncommutativity should be

introduce to some two specific space-time coordinates. Let us suppose that the specified

NC coordinates are denoted by xi and xj (i < j) which satisfies [xi, xj]? = iθ.

First, let us comment on an important formula which is relevant to one-soliton solu-

tions. Defining new coordinates z := xi + vxj, z := xi − vxj, we can easily see

f(z) ? g(z) = f(z)g(z) (5.1)

because the Moyal-product (2.2) is rewritten in terms of (z, z) as [49]

f(z, z) ? g(z, z) = eivθ(∂z′∂z′′−∂z′∂z′′ )f(z′, z′)g(z′′, z′′)∣∣∣ z′ = z′′ = z

z′ = z′′ = z.

(5.2)

14

Hence NC one soliton-solutions are essentially the same as commutative ones for both

space-time and space-space noncommutativity cases.

When f(x) is a linear function, the treatment of ef(x)? becomes easy as follows:

(eξ(~x;α)? )−1 = e−ξ(~x;α)

? , (5.3)

∂xeξ(~x;α)? = αeξ(~x;α)

? . (5.4)

The proof can be seen as well from the fact that because of Eq. (5.2), the star exponential

function of a linear function itself reduces to commutative one, that is, eξ(~x,α)? = eξ(~x,α).

These formula are crucial in discussion on asymptotic behavior of N -soliton solutions.

Furthermore, the Baker-Campbell-Hausdorff (BCH) formula implies

eξ(~x;α)? ? eξ(~x;β)

? = e(i/2)θ(αiβj−αjβi)eξ(~x;α)+ξ(~x;β)? . = eiθ(αiβj−αjβi)eξ(~x;β)

? ? eξ(~x;α)? . (5.5)

The factor (i/2)θ(αiβj−αjβi) can be absorbed by a coordinate shift in ξ(~x; α), and hence

there is a possibility that noncommutativity might affect coordinate shifts by the factor

such as phase shifts in the asymptotic behavior. When coordinates and fields are treated

as complex, such a coordinate shift by a complex number causes no problem. However,

if we want to apply NC integrable equations to real phenomena, such as, shallow water

waves, then it becomes hard to interpret physically. Let us see what happens in the

asymptotic region.

5.1 Asymptotic behavior of NC KdV hierarchy

First, let us discuss the NC KdV hierarchy and the asymptotic behavior of the N -soliton

solutions. The NC KdV hierarchy is the 2-reduction of the NC KP hierarchy and re-

alized by putting βs = −αs on the N -soliton solutions of the NC KP hierarchy. Here

the constants αs and as are non-zero real numbers and as is positive. Because of the

permutation property of the columns of quasi-determinants in propositon 3.1 (i), we can

assume α1 < α2 < · · · < αN .

In the NC KdV hierarchy, the x2n-th flow becomes trivial and in the x2n+1-th flow

equation, space and time coordinates are specified as (x, t) ≡ (x1, x2n+1).

Now let us define a new coordinate x := x + α2nI t comoving with the I-th soliton and

take t → ±∞ limit.5 Then, because of x+α2ns t = x+α2n

I t+(α2ns −α2n

I )t, either eαs(x+α2n

s t)?

5Such kind of observation for soliton scatterings in NC integrable equations is first seen in [50]. (Seealso [2].)

15

or e−αs(x+α2n

s t)? goes to zero for s 6= I. Hence the behavior of fs becomes at t → +∞:

fs(~x) −→

ase−αs(x+α2n

s t)? s < I

eαI(x+α2n

I t)? + aIe

−αI(x+α2nI t)

? s = I

eαs(x+α2n

s t)? s > I,

(5.6)

and at t → −∞:

fs(~x) −→

eαs(x+α2n

s t)? s < I

eαI(x+α2n

I t)? + aIe

−αI(x+α2nI t)

? s = I

ase−αs(x+α2n

s t)? s > I.

(5.7)

We note that the s-th (s 6= I) column is proportional to a single exponential function

e±αs(x+α2

st)? due to Eq. (5.4). Because of the multiplication property of columns of quasi-

determinants in propositon 3.1 (ii), we can eliminate a common invertible factor from

the s-th column in |A|ij where s 6= j. (Note that this exponential function is actually

invertible as is shown in Eq. (5.3).) Hence the N -soliton solution becomes the following

simple form where only the I-th column is non-trivial, at t → +∞:

ΦN ? f →1 · · · 1 e

ξ(~x;αI)? + aIe

−ξ(~x;αI)? 1 · · · 1 f

−α1 · · · −αI−1 αI(eξ(~x;αI)? − aIe

−ξ(~x;αI)? ) αI+1 · · · αN f ′

......

......

......

(−α1)N−1 · · · (−αI−1)

N−1 αN−1I (e

ξ(~x;αI)? + (−1)N−1aIe

−ξ(~x;αI)? ) αN−1

I+1 · · · αN−1N f (N−1)

(−α1)N · · · (αI−1)

N αNI (e

ξ(~x;αI)? + (−1)NaIe

−ξ(~x;αI)? ) αN

I+1 · · · αNN f (N)

,

and at t → −∞:

ΦN ? f →1 · · · 1 e

ξ(~x;αI)? + aIe

−ξ(~x;αI)? 1 · · · 1 f

α1 · · · αI−1 αI(eξ(~x;αI)? − aIe

−ξ(~x;αI)? ) −αI+1 · · · −αN f ′

......

......

......

αN−11 · · · αN−1

I−1 αN−1I (e

ξ(~x;αI)? + (−1)N−1aIe

−ξ(~x;αI)? ) (−αI+1)

N−1 · · · (−αN)N−1 f (N−1)

αN1 · · · αN

I−1 αNI (e

ξ(~x;αI)? + (−1)NaIe

−ξ(~x;αI)? ) (−αI+1)

N · · · (−αN)N f (N)

.

Here we can see that all elements in between the first column and the N -th column

commute and depend only on x + α2nI t in ξ(~x; αI), which implies that the corresponding

asymptotic configuration coincides with the commutative one,6 that is, the I-th one-

soliton configuration with some coordinate shift so called the phase shift. The commuta-

tive discussion has been studied in this way by many authors, and therefore, we conclude6Note that because f is arbitrary, there is no need to consider the products between a column and

the (N + 1)-th column. This observation for asymptotic behavior can be made from Eq. (4.37) also.

16

that for the NC KdV hierarchy, asymptotic behavior of the multi-soliton solutions is all

the same as commutative one, and as the results, the N-soliton solutions possess N lo-

calized energy densities and in the scattering process, they never decay and preserve their

shapes and velocities of the localized solitary waves. The phase shifts also occur by the

same degree as commutative ones.

Finally, we make a brief comment on the 2-soliton solutions. In this situation, space-

time dependence appears only as two kind of exponential factors e±α1(x+α2n1 t) and e±α2(x+α2n

2 t).

Noncommutativity of them could have effects by the factor e±(i/2)α1α2(α2n1 −α2n

2 )θ because of

the BCH formula. However, if the two kind of parameters satisfy

1

2α1α2(α

2n1 − α2n

2 )θ = 2πk, (5.8)

where k is an non-zero integer, then, the effects of noncommutativity perfectly disappear

at the every stage of calculations and the behavior of the 2-soliton solution perfectly

coincides with that of commutative one at any time and any location. However the

condition (5.8) is given specially by hand, and the mathematical and physical meaning of

this observation is still unknown.

5.2 Asymptotic behavior of NC KP hierarchy

Now, let us discuss the NC KP hierarchy and the asymptotic behavior of the N -soliton

solutions. The space and time coordinates are (x, y, t) ≡ (x1, x2, xn) and noncommuta-

tivity is introduced into some specified two coordinates among x, y and t. The specified

NC coordinates are also denoted by xi and xj with [xi, xj]? = iθ. Here the constants αs

and βs are non-zero real numbers and the constant as will be redefined later.

As we mentioned at the beginning of the present section, one-soliton solutions are

all the same as commutative ones. However, we have to treat carefully for the NC KP

hierarchy. From Eq. (4.37), naive one-soliton solution can be expressed as follows

u2 = ∂x

(∂x(e

ξ(~x;α)? + aeξ(~x;β)

? ) ? (eξ(~x;α)? + aeξ(~x;β)

? )−1)

= ∂x

((αi + aβi∆eη(~x;α,β)

? ) ? (1 + a∆eη(~x;α,β)? )−1

), (5.9)

where

η(~x; α, β) := x(β − α) + y(β2 − α2) + t(βn − αn)

∆ := ei2θ(αiβj−αjβi). (5.10)

We note that the factor ∆ can be absorbed by redefining a coordinate such as x →x + (β − α)−1(i/2)θ(αiβj − αjβi). The final form of the solution depend only on xi(β

iI −

17

αiI) + xj(β

jI − αj

I) for NC coordinates and the Moyal products disappear. Hence there

becomes no dependence of complex numbers, and the one-soliton solution is the same

as commutative one in this sense. However now we treat the coordinates as real and it

would be better to redefine a positive real number a which satisfies a = a∆−1, so that

f1 = eξ(~x;α)? + ae

ξ(~x;β)? =

(1 + ae

η(~x;α,β)?

)? e

ξ(~x;α)? , in order to avoid such a coordinate shift

by a complex number.

This point becomes important for the multi-soliton solutions. The constants as in

the N -soliton solution of the NC KP hierarchy should be replaced with a positive real

number as which satisfies as = as∆−1s where ∆s := e(i/2)θ(αi

sβjs−αj

sβis), because the N -soliton

configuration reduces to a (I-th) one-soliton configuration when we set αs = βs = 0 for

all s(6= I).

Let us define new coordinates comoving with the I-th soliton as follows:

p := x + αIy + αn−1I t, q := x + βIy + βn−1

I t. (5.11)

Then the function ξ(x, y, t; αs) can be rewritten in terms of the new coordinates as

ξ(p, q, xr; αs) = A(αs)p + B(αs)q + C(αs)xr where xr is a specified coordinate among

x, y and t, and A(αs), B(αs) and C(αs) are real constants depending on αI , βI and αs.

For example, in the case of xr ≡ t, we can get from Eq. (5.11)

(xy

)=

1

βI − αI

(βIp− αIq + αIβI(β

n−2I − αn−2

I )t−p + q + (αn−1

I − βn−1I )t

), (5.12)

and find

ξ = x + αsy + αn−1s t

=βI − αs

βI − αI

p +αs − αI

βI − αI

q +

{αn−1

s +αIβI(β

n−2I − αn−2

I ) + αs(αn−1I − βn−1

I )

βI − αI

}t.

Here we suppose that C(αs) 6= C(βs) which corresponds to pure soliton scatterings.7

Now let us take xr → ±∞ limit, then, for the same reason as in the NC KdV hierarchy,

we can see that the asymptotic behavior of fs becomes:

fs(~x) −→{

Aseξ(~x;γs)? s 6= I

eξ(~x;αI)? + aIe

ξ(~x;βI)? s = I

(5.13)

where As is some real constant whose value is 1 or as, and γs is a real constant taking a

value of αs or βs. As in the case of the NC KdV hierarchy, the s-th (s 6= I) column is

proportional to a single exponential function and we can eliminate this factor from the

7The condition C(αs) = C(βs) could lead to soliton resonances in commutative case.

18

s-th column. Hence in the asymptotic region xr → ±∞, the N -soliton solution becomes

the following simple form where only the I-th column is non-trivial:

ΦN ? f →

1 · · · 1 eξ(~x;αI)? + aIe

ξ(~x;βI)? 1 · · · 1 f

γ1 · · · γI−1 αIeξ(~x;αI)? + aIβIe

ξ(~x;βI)? γI+1 · · · γN f ′

......

......

......

γN−11 · · · γN−1

I−1 αN−1I e

ξ(~x;αI)? + aIβ

N−1I e

ξ(~x;βI)? γN−1

I+1 · · · γN−1N f (N−1)

γN1 · · · γN

I−1 αNI e

ξ(~x;αI)? + aIβ

NI e

ξ(~x;βI)? γN

I+1 · · · γNN f (N)

=

1 · · · 1 1 + aIeη(~x;αI ,βI)? 1 · · · 1 f

γ1 · · · γI−1 αI + aIβIeη(~x;αI ,βI)? γI+1 · · · γN f ′

......

......

......

γN−11 · · · γN−1

I−1 αN−1I + aIβ

N−1I e

η(~x;αI ,βI)? γN+1

I+1 · · · γN−1N f (N−1)

γN1 · · · γN

I−1 αNI + aIβ

NI e

η(~x;αI ,βI)? γN

I+1 · · · γNN f (N)

.

Here we can see that all elements between the first column and the N -th column commute

and depend only on xi(βiI −αi

I) + xj(βjI −αj

I) for NC coordinates, which implies that the

corresponding asymptotic configuration coincides with the commutative one. Hence, we

can also conclude that for the NC KP hierarchy, asymptotic behavior of the multi-soliton

solutions is all the same as commutative one in the process of pure soliton scatterings, and

as the results, the N-soliton solutions possess N localized energy densities and in the pure

scattering process, they never decay and preserve their shapes and velocities of localized

solitary waves. Asymptotic behavior of two-soliton solution of NC KP equation studied

by Paniak [23] actually coincides with our result for n = 3, N = 2.

Now we restricted ourselves to the NC KP hierarchy, however, this observation would

be also true of other kind of NC hierarchies, such as, the (2-dimensional) NC Toda lattice

hierarchy [51], the NC toroidal GD hierarchy presented in section 4 and the NC matrix

KP hierarchy [52] because the soliton solutions could be represented by such kind of

(generalized) Wronski matrices here, and the asymptotic analysis would be almost the

same.

6 Conservation Laws for NC KP Hierarchy

Here we prove the existence of infinite conservation laws for the wide class of NC soliton

equations. The existence of infinite number of conserved quantities would lead to infinite-

dimensional hidden symmetry from Noether’s theorem.

First we would like to comment on conservation laws of NC field equations. The

discussion is basically the same as commutative case because both the differentiation and

19

the integration are the same as commutative ones in the Moyal representation.

Let us suppose the conservation law

∂tσ(t, xi) = ∂iJi(t, xi), (6.1)

where space and time coordinates are denoted by xi and t, respectively, and σ(t, xi)

and J i(t, xi) are called the conserved density and the associated flux, respectively. The

conserved quantity is given by spatial integral of the conserved density:

Q(t) =∫

spacedDxσ(t, xi), (6.2)

where the integral∫space dxD is taken for spatial coordinates. The proof is straightforward:

dQ

dt=

∂

∂t

∫

spacedDxσ(t, xi) =

∫

spacedDx∂iJi(t, xi)

=∫

spatialinfinity

dSiJi(t, xi) = 0, (6.3)

unless the surface term of the integrand Ji(t, xi) vanishes. The convergence of the integral

is also expected because the star-product naively reduces to the ordinary product at spatial

infinity due to: ∂i ∼ O(r−1) where r := |x|.Here let us return back to the NC KP hierarchy. In order to discuss the conservation

laws, we have to specify for what equations the conservation laws are. The specified

equations possess the corresponding space-time coordinates in the infinite coordinates

x1, x2, x3, · · ·. Identifying t ≡ xm, we can get infinite conserved densities for the NC

hierarchies as follows (n = 1, 2, . . .) [36]:

σn = res−1Ln + θim

m−1∑

k=0

k∑

l=0

(kl

)∂k−l

x res−(l+1)Ln ¦ ∂ireskL

m, (6.4)

where the suffices i must run in the space-time directions only. The product “¦” is called

Strachan’s product [53] and defined by

f(x) ¦ g(x) :=∞∑

p=0

(−1)p

(2p + 1)!

(1

2θij∂

(x′)i ∂

(x′′)j

)2p

f(x′)g(x′′)∣∣∣x′=x′′=x

. (6.5)

This is a commutative and non-associative product.

We can easily see that deformation terms appear in the second term of Eq. (6.4) in

the case of space-time noncommutativity. On the other hand, in the case of space-space

noncommutativity, the conserved density is given by the residue of Ln as commutative

case.

20

For examples, explicit representation of the NC KP equation with space-time noncom-

mutativity, the NC KdV equation is

σn = res−1Ln − 3θ ((res−1L

n) ¦ u′3 + (res−2Ln) ¦ u′2) . (6.6)

We make a comment that conserved densities for one-soliton configuration are not

deformed in the NC extension because one soliton solutions can be always reduced to

commutative ones

7 Conclusion and Discussion

In this paper, we studied exact multi-soliton solutions of NC integrable hierarchies, includ-

ing NC KP and toroidal KP hierarchies and the reductions, in terms of quasi-determinants.

We found that the asymptotic behavior of them could be all the same as commutative ones

in the process of (pure) soliton scatterings. This implies that the exact soliton solutions

are actually solitons in the sense that the configuration has localized energy densities and

never decay, and the phase shifts also appear by the same degree as in the commutative

case.

It would be reasonable that there is no difference in asymptotic behavior of pure soli-

ton scatterings on between commutative and NC spaces, because in asymptotic region,

star-products reduce to ordinary commutative products and the effect of noncommuta-

tivity disappears. These results imply that we cannot detect effects of noncommutativity

of space-time by observing such soliton dynamics. However, total behavior of them is

unknown and it is worth studying further to find different aspects of the NC soliton

dynamics from commutative ones.

Dynamics in soliton resonances is also interesting. From the present results of pure

soliton scatterings, we could naturally expect that the configurations in soliton resonances

would not be affected by noncommutativity in asymptotic region though we might need

to make further modifications in the multi-soliton solutions such as as = as∆−1s for the

NC KP equation. Quantum treatments of the soliton scatterings is also interesting, such

as properties of factorized S-matrix of NC sine-Gordon model [54]. Furthermore, the

existence of multi-soliton solutions is important in integrable systems and the present

observations might be a hint to reveal NC Hirota’s bilinearization, theory of tau-functions

and the structure of solution spaces.

21

Acknowledgments

The author would like to thank C. S. Chu, H. Kajiura, L. Mason, S. Moriyama, Y. Ohta,

T. Suzuki, K. Takasaki and K. Toda for useful comments. He is also grateful to C. R. Gilson,

J. J. C. Nimmo and I. Strachan for warm hospitality and fruitful discussion during stay at

the department of mathematics, university of Glasgow. This work was partially supported

by the Daiko foundation (#9095) and the Yamada Science Foundation for the promotion

of the natural science, and Grant-in-Aid for Young Scientists (#18740142).

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