On a direct approach to quasideterminant solutions of a noncommutative KP equation

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by  C. R. Gilson

On a direct approach to quasideterminant solutions of a
noncommutative KP equation
C. R. Gilson and J. J. C. Nimmo
Department of Mathematics,
University of Glasgow
Glasgow G12 8QW, UK
7 Abstract
0
A noncommutative version of the KP equation and two families of its solutions expressed as
0
quasideterminants are discussed. The origin of these solutions is explained by means of Darboux
2
and binary Darboux transformations. Additionally, it is shown that these solutions may also be

n verifieddirectly. ThisapproachisreminiscentofthewronskiantechniqueusedfortheHirotabilinear
a
form of the regular, commutative KP equation but, in the noncommutative case, no bilinearising
J
transformation is available.
6
1
1 Introduction

]
I
S
There has been recent interest in a noncommutative version of the Kadomtsev-Petviashvili equation
.
n (ncKP) [1, 2, 3, 4, 5, 6, 7, 8]
i
l
n (v +v +3v v ) +3v −3[v ,v ]=0. (1)
t xxx x x x yy x y
[

This equation can be obtained via the compatibility of the same Lax pair (7)–(8) as is used in the
2
v commutative case, but it is not assumed that v and its derivatives commute. In the case that variables
7 do commute, we may differentiate (1) with respect to x and set v = u to obtain the well-known
x
2 (commuting) KP equation
0
(u +u +6uu ) +3u =0. (2)
1 t xxx x x yy
0 In most of the recent work on ncKP the noncommutativity arises because of a quantization of the
7
phase space in which independent variables do not commute and the (commutative) product of real-
0
or complex-valued functions of these are replaced by the associative but noncommutative Moyal star
/
n product. Thisapproachisusefulfromthepointofviewofinterpretingsolutionsastheycanbeexpressed
i
l intermsofstandardfunctions. Itishoweverconceptuallyquitedifficultbecauseofthenoncommutativity
n
of the independent variables.
:
v In this present paper we will not specify the nature of the noncommutativity and the results we
i
X present are valid not only in the star product case but also for, for example, the matrix or quarternion
versionsoftheKPequation. ThisisinthespiritoftheworkbyEtingof,GelfandandRetakh[9]inwhich
r
a solutionsofthencKPequationwerefoundintermsofquasideterminants[10,11]usinganoncommutative
versionof Gelfand-Dickey theory. Very recently, Hamanaka [12] has used this form of solution to obtain
the soliton solutions of ncKP in the Moyal product case.
We will consider two types of quasideterminant solutions of ncKP. One is equivalent to those found
in [9] which we will call quasiwronskians. We will also consider a new type of quasideterminant solution
whichwetermaquasigrammian. These twotypesofsolutionareeachconstructedbyiteratingDarboux
transformations;thequasiwronskiansusingastandardDarbouxtransformationandthequasigrammians
usingthe relatedbinaryDarbouxtransformation. The connectionbetweenDarbouxtransformationsfor
a matrix Schr¨odinger equation and quasideterminants was also investigated in [13].
We will then show that, in fact, these solutions can be verified by direct substitution. This sort
of direct approach is very widely studied in the commutative case (see [14] for the first results for the
KP case, and [15] for a discussion of many other examples). In these cases one first makes a change of
1
dependent variable which converts the nonlinear equation to Hirota bilinear form. For the KP equation
onewritesu=2(logτ) andthen(2)isconvertedtoHirotaform,ahomogeneouslyquadraticdifferential
xx
equation in τ. A solution τ in the form of a determinant may be verifiedby recognisingthe Hirota form
as a determinantal identity such as a Plu¨cker relation or Jacobi identity.
In contrast, the central role of the τ-function (a determinant) in the commutative case is taken
by the quasideterminant in the noncommutative case. In this case there is no bilinearising change of
variables since v is expressed directly as a quasideterminant. Also, remarkably given what happens in
the commutative case, no use is made of special identities in verifying the quasideterminant solutions of
ncKP.Some remarksonthe reasonfor this are givenlater. Paradoxically,directverificationofsolutions
in the noncommutative case is in a number of respects easier than in the commutative case. However,
this state of affairs seems to be particular to ncKP. In other examples we have considered [16, 17, 18] a
change of variables and use of quasideterminant identities are necessary to achieve direct verification of
solutions.
2 Preliminaries
In this short section we recall some of the key elementary properties of quasideterminants. The reader
is referred to the original papers [10, 11] for a more detailed and general treatment.
2.1 Quasideterminants
An n×n matrix A over a ring R (non-commutative, in general) has n2 quasideterminants written as
|A| for i,j =1,...,n, which are also elements of R. They are defined recursively by
i,j
|A| =a −rj(Ai,j)−1ci, A−1 =(|A|−1) .
i,j i,j i j j,i i,j=1,...,n
In the above rj represents the ith row of A with the jth element removed, ci the jth column with the
i j
ithelement removedandAi,j the submatrixobtainedby removingtheith rowandthejth columnfrom
A. Quasideterminants can also denoted as shown below by boxing the entry about which the expansion
is made
Ai,j ci
j
|A|i,j = rj a .
(cid:12) i i,j (cid:12)
(cid:12) (cid:12)
(cid:12) (cid:12)
The case n = 1 is rather trivial; let A = (a),(cid:12)say, and th(cid:12)en there is one quasideterminant |A|1,1 =
(cid:12) (cid:12)
a b
| a |=a. For n=2, let A= , then there are four quasideterminants
c d
(cid:18) (cid:19)
a b a b
|A| = =a−bd−1c, |A| = =b−ac−1d,
1,1 c d 1,2 c d
(cid:12) (cid:12) (cid:12) (cid:12)
(cid:12) (cid:12) (cid:12) (cid:12)
(cid:12)a b(cid:12) (cid:12)a b (cid:12)
|A|2,1 =(cid:12) c d(cid:12)=c−db−1a, |A|2,2 = (cid:12)c d (cid:12)=d−ca−1b.
(cid:12) (cid:12) (cid:12) (cid:12)
(cid:12) (cid:12) (cid:12) (cid:12)
From this we can obtain the m(cid:12) atrix(cid:12)inverse, (cid:12) (cid:12)
(cid:12) (cid:12) (cid:12) (cid:12)
(a−bd−1c)−1 (c−db−1a)−1
A−1 = ,
(b−ac−1d)−1 (d−ca−1b)−1
(cid:18) (cid:19)
whichis thenusedinthe definitionofthe 9 quasideterminantsofa 3×3matrix. Note that ifthe entries
in A commmute, the above becomes the familiar formula for the inverse of a 2×2 matrix with entries
expressed as ratios of determinants. Indeed this is true for any size of square matrix; if the entries in A
commute then
det(A)
|A| =(−1)i+j . (3)
i,j det(Ai,j)
In this paper we will consider only quasideterminants that are expanded about a term in the last
column, most usually the last entry. For a block matrix
A B
C d
(cid:18) (cid:19)
2
where d ∈ R, A is a square matrix over R of arbitrary size and B, C are column and row vectors over
R of compatible lengths, we have
A B
=d−CA−1B.
C d
(cid:12) (cid:12)
(cid:12) (cid:12)
(cid:12) (cid:12)
2.2 Invariance under row an(cid:12)d colu(cid:12)mn operations
The quasideterminants of a matrix have invariance properties similar to those of determinants under
elementary row and column operations applied to the matrix. Consider the following quasideterminant
of an n×n matrix;
E 0 A B EA EB A B
= =g(d−CA−1B)=g . (4)
F g C d FA+gC FB+gd C d
(cid:12)(cid:18) (cid:19)(cid:18) (cid:19)(cid:12)n,n (cid:12) (cid:12)n,n (cid:12) (cid:12)n,n
(cid:12) (cid:12) (cid:12) (cid:12) (cid:12) (cid:12)
The above(cid:12) formula can be u(cid:12)sed to(cid:12)understand the effect(cid:12) on a quasideterminant o(cid:12)f certa(cid:12)in elementary
(cid:12) (cid:12) (cid:12) (cid:12) (cid:12) (cid:12)
row operations involving multiplication on the left. This formula excludes those operations which add
left-multiples of the row containing the expansion point to other rows since there is no simple way to
describe the effect of these operations. For the allowed operations however, the results can be easily
described; left-multiplying the row containing the expansionpoint by g has the effect of left-multiplying
the quasideterminant by g and all other operations leave the quasideterminant unchanged. There is
analogous invariance under column operations involving multiplication on the right.
2.3 Noncommutative Jacobi Identity
There is a quasideterminant version of Jacobi’s identity for determinants, called the noncommutative
Sylvester’s Theorem by Gelfand and Retakh [10]. The simplest version of this identity is given by
A B C −1
A C A B A B A C
D f g = − . (5)
3 Solutions ob(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)Etaihnedi (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)us(cid:12)(cid:12)(cid:12)(cid:12)Eingi (cid:12)(cid:12)(cid:12)(cid:12)Da(cid:12)(cid:12)(cid:12)(cid:12)Erbohu(cid:12)(cid:12)(cid:12)(cid:12)x(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)Dtrfan(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) sf(cid:12)(cid:12)(cid:12)(cid:12)Dormga(cid:12)(cid:12)(cid:12)(cid:12)tions
3.1 Darboux transformation
Let L be an operator covariant under the Darboux transformation G = θ∂ θ−1 = ∂ −θ θ−1 where
θ x x x
θ is an eigenfunction of L (i.e. L(θ) = 0). Let θ , i = 1,...,n be a particular set of eigenfunctions
i
(i−1)
and introduce the notation Θ = (θ ,...,θ ) and Θ = (θ ) , the n×n wronskian matrix of
1 n j i,j=1,...,n
θ ,...,θ , where (k) denotes the kth x-derivative.
1 n
Let φ = φ be a general eigenfunction of Lb = L and θ = θ . Then φ := G (φ ) and
[1] [1] [1] 1 [2] θ[1] [1]
θ =φ | are eigenfunctions for L =G L G−1. In general,for n≥1 define the nth Darboux
[2] [2] φ→θ2 [2] θ[1] [1] θ[1]
transform of φ by
φ =φ(1)−θ(1)θ−1φ ,
[n+1] [n] [n] [n] [n]
in which
θ = φ .
[k] [k] φ→θk
It is known that [13, 11] (cid:12)
(cid:12)
Θ φ
. .
. .
(cid:12) . . (cid:12)
φ =(cid:12) (cid:12). (6)
[n+1] (cid:12)Θ(n−1) φ(n−1)(cid:12)
(cid:12) (cid:12)
(cid:12)(cid:12) Θ(n) φ(n) (cid:12)(cid:12)
(cid:12) (cid:12)
(cid:12) (cid:12)
The Lax pair for the ncKP equation (1) is (cid:12) (cid:12)
(cid:12) (cid:12)
L=∂2+v −∂ , (7)
x x y
M =4∂3+6v ∂ +3v +3v +∂ . (8)
x x x xx y t
3
BothLandM arecovariantwithrespecttotheaboveDarbouxtransformation. Moreover,itisstraight-
forward to calculate that the effect of the Darboux transformation
L˜ =G LG−1, M˜ =G MG−1,
θ θ θ θ
is that v˜=v+2θ θ−1. Thus after n Darboux transformations we obtain
x
n
v =v+2 θ θ−1, (9)
[n+1] [i],x [i]
i=1
X
which describes a class of solutions of ncKP. An analogousformula is obtained using a noncommutative
versionof Gelfand-Dickey theory in [9]. Further, it may be proved by induction from (6), making use of
an identity of the form (5), that
Θ 0
. .
. .
n (cid:12) . . (cid:12)
θ[i],xθ[−i]1 =−(cid:12)(cid:12)Θ(n−2) 0 (cid:12)(cid:12). (10)
(cid:12) (cid:12)
Xi=1 (cid:12)Θ(n−1) 1 (cid:12)
(cid:12) (cid:12)
(cid:12) Θ(n) 0 (cid:12)
(cid:12) (cid:12)
(cid:12) (cid:12)
(cid:12) (cid:12)
Thus,usingDarbouxtransformations,wehaveobtain(cid:12)edaformula(cid:12)forsolutionsv[n+1] ofncKPexpressed
in terms of a known solution v and a single wronskian-like quasideterminant,
Θ 0
. .
. .
(cid:12) . . (cid:12)
v[n+1] =v−2(cid:12)(cid:12)Θ(n−2) 0 (cid:12)(cid:12). (11)
(cid:12) (cid:12)
(cid:12)Θ(n−1) 1 (cid:12)
(cid:12) (cid:12)
(cid:12) Θ(n) 0 (cid:12)
(cid:12) (cid:12)
(cid:12) (cid:12)
(cid:12) (cid:12)
3.2 Binary Darboux transformation (cid:12) (cid:12)
To define a binary Darboux transformation one needs to consider the adjoint Lax pair. The notion of
adjoint is easily extended from the familiar matrix situation to any ring R (see [19]); suppose that for
each a∈R, there exists a† ∈R, for any derivative ∂ acting on R, ∂† =−∂ and for any product AB of
elements of, or operators on R, (AB)† =B†A†. Accordingly, the adjoint Lax pair is
L† =∂2+v† +∂ , (12)
x x y
M† =−4∂3−6v†∂ −3v† +3v†−∂ . (13)
x x x xx y t
FollowingthestandardconstructionofabinaryDarbouxtransformation(see[20,21])oneintroduces
a potential Ω(φ,ψ) satisfying
Ω(φ,ψ) =ψ†φ, Ω(φ,ψ) =ψ†φ −ψ†φ, Ω(φ,ψ) =−4(ψ†φ −ψ†φ +ψ† φ)−6ψ†v φ. (14)
x y x x t xx x x xx x
The parts of this definition are compatible when L(φ) = M(φ) = 0 and L†(ψ) = M†(ψ) = 0. More
generally,we candefine Ω(Φ,Ψ) for any row vectorsΦ and Ψ such that L(Φ)=M(Φ)=0 and L†(Ψ)=
M†(Ψ)=0. If Φ is an n-vector and Ψ is an m-vector then Ω(Φ,Ψ) is an m×n matrix. The adjoint of
a p×q matrix A=(a ) over R has an obvious meaning. It is the q×p matrix A† =(a† ).
i,j j,i
A binary Darboux transformation is then defined by
φ =φ −θ Ω(θ ,ρ )−1Ω(φ ,ρ )
[n+1] [n] [n] [n] [n] [n] [n]
and
ψ =ψ −ρ Ω(θ ,ρ )−†Ω(θ ,ψ )†,
[n+1] [n] [n] [n] [n] [n] [n]
where
θ = φ , ρ = ψ .
[n] [n] φ→θn [n] [n] ψ→ρn
(cid:12) (cid:12)
(cid:12) (cid:12)
4
Using the notation Θ=(θ ,...,θ ) and P=(ρ ,...,ρ ) it is easy to proveby induction that for n≥1,
1 n 1 n
Ω(Θ,P) Ω(φ,P)
φ = , (15)
[n+1] Θ φ
(cid:12) (cid:12)
(cid:12) (cid:12)
(cid:12) (cid:12)
(cid:12) (cid:12)
Ω(cid:12)(Θ,P)† Ω(Θ,ψ)(cid:12)†
ψ = , (16)
[n+1] P ψ
(cid:12) (cid:12)
(cid:12) (cid:12)
(cid:12) (cid:12)
and (cid:12) (cid:12)
(cid:12) Ω(Θ,P) Ω(φ(cid:12),P)
Ω(φ ,ψ )= . (17)
[n+1] [n+1] Ω(Θ,ψ) Ω(φ,ψ)
(cid:12) (cid:12)
(cid:12) (cid:12)
(cid:12) (cid:12)
As for the effect of these transformation on the Lax(cid:12) pair, a transforma(cid:12)tion by θ,ρ gives new coefficients
(cid:12) (cid:12)
defined in terms of
vˆ=v+2θΩ(θ,ρ)−1ρ†.
Thus after n binary Darboux transformations we obtain
n
v =v+2 θ Ω(θ ,ρ )−1ρ† , (18)
[n+1] [k] [k] [k] [k]
k=1
X
and this may be reexpressed in terms of a single quasideterminant as
Ω(Θ,P) P†
v =v−2 . (19)
[n+1] Θ 0
(cid:12) (cid:12)
(cid:12) (cid:12)
(cid:12) (cid:12)
In this way we have obtained a second expressionfo(cid:12)r solutions of(cid:12)the ncKP equation, this time in terms
of grammian-type quasideterminants.
In the following sections, we will show that these two quasideterminant solutions (wronskian-type
and grammian-type) of the ncKP equation may also be verified by direct calculation in the spirit of
Hirota’s direct method.
4 Derivatives of a quasideterminant
We can derive a rather appealing formula for derivatives of a quasideterminant which resembles the
formula for derivatives of a normal determinant. Consider the derivative
′
A B
=d′−C′A−1B+CA−1A′A−1B−CA−1B′ (20)
C d
(cid:12) (cid:12)
(cid:12) (cid:12)
(cid:12) (cid:12)
where A is an n × n ma(cid:12)trix, C(cid:12) is a row vector and B a column vector. If the matrix A has the
grammian-like property that its derivative is a scalar product
k
A′ = E F ,
i i
i=1
X
where E (F ) are column (row) vectors of appropriate length, then the third term on the RHS of (20)
i i
can be factorised as a product of quasideterminants, i.e.
′ k
A B
=d′−C′A−1B+ (CA−1E )(F A−1B)−CA−1B′
i i
C d
(cid:12) (cid:12) i=1
(cid:12) (cid:12) X
(cid:12)(cid:12) (cid:12)(cid:12) A B A B′ k A Ei A B
= + + . (21)
(cid:12)C′ d′ (cid:12) (cid:12)C 0 (cid:12) i=1(cid:12)C 0 (cid:12)(cid:12)Fi 0 (cid:12)
(cid:12) (cid:12) (cid:12) (cid:12) X(cid:12) (cid:12)(cid:12) (cid:12)
(cid:12) (cid:12) (cid:12) (cid:12) (cid:12) (cid:12)(cid:12) (cid:12)
(cid:12) (cid:12) (cid:12) (cid:12) (cid:12) (cid:12)(cid:12) (cid:12)
5
Evenifthe matrixAdoes nothavethis grammian-likestructurethen the thirdtermonthe RHSof(20)
can still be factorized as a product by inserting the n×n identity matrix expressed in the form
n
I = e et,
k k
k=1
X
wheree isthen-vector(δ )(i.e.acolumnvectorwith1inthekthrowand0elsewhere). LetZk denote
k ik
the kth row and Z the kth column of a matrix Z. In this way we have
k
′ n n
A B
=d′−C′A−1B+ (CA−1e )(etA′A−1B)− (CA−1e )etB′.
C d k k k k
(cid:12) (cid:12) k=1 k=1
(cid:12) (cid:12) X X
(cid:12) (cid:12)
Note here that(cid:12) we hav(cid:12)e also introduced this form of the identity into the last term on the RHS. This
gives
A B ′ A B n A e A B
k
= + . (22)
In a similar way, by inser(cid:12)(cid:12)(cid:12)(cid:12)tCing tdhe(cid:12)(cid:12)(cid:12)(cid:12)iden(cid:12)(cid:12)(cid:12)(cid:12)tCit′y mda′t(cid:12)(cid:12)(cid:12)(cid:12)rix kiXn=1a(cid:12)(cid:12)(cid:12)(cid:12)Cdiffer0en(cid:12)(cid:12)(cid:12)(cid:12)t(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(pAoks)it′ion(wBek)h′a(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)ve a column versionof the
derivative formula
A B ′ A B′ n A (Ak)′ A B
= + . (23)
4.1 Derivatives of q(cid:12)(cid:12)(cid:12)(cid:12)Cuasdiw(cid:12)(cid:12)(cid:12)(cid:12)ron(cid:12)(cid:12)(cid:12)(cid:12)Cskiadn′ (cid:12)(cid:12)(cid:12)(cid:12)s kX=1(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)C (Ck)′ (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)etk 0 (cid:12)(cid:12)(cid:12)(cid:12)
In this section we will calculate derivatives of a quasideterminant of the form
Θ e
Q(i,j)= n−j , (24)
(cid:12)Θ(n+i) 0 (cid:12)
(cid:12) b (cid:12)
(cid:12) (cid:12)
where, as above, Θ = (θj(i−1))i,j=1,...,n is the n×(cid:12)(cid:12) n wronskian(cid:12)(cid:12)matrix of θ1,...,θn. In this definition,
i and j are allowed to take any integer values subject to the convention that if n − j lies outside
the range 1,2,...b,n, then en−j = 0 and so Q(i,j) = 0. There is an important special case; when
n+i=n−j−1∈[0,n−1], (i.e. i+j+1=0 and −n≤i<0) we have
Θ 0 Θ 0
. . . .
. . . .
(cid:12) . . (cid:12) (cid:12) . . (cid:12)
(cid:12)(cid:12)Θ(n+i) 1 (cid:12)(cid:12) (cid:12)(cid:12)Θ(n+i) 1 (cid:12)(cid:12)
Q(i,j)=(cid:12) (cid:12)=(cid:12) (cid:12)=−1,
(cid:12) .. .. (cid:12) (cid:12) .. .. (cid:12)
(cid:12) . . (cid:12) (cid:12) . . (cid:12)
(cid:12) (cid:12) (cid:12) (cid:12)
(cid:12)Θ(n−1) 0 (cid:12) (cid:12)Θ(n−1) 0 (cid:12)
(cid:12) (cid:12) (cid:12) (cid:12)
(cid:12)Θ(n+i) 0 (cid:12) (cid:12) 0 -1 (cid:12)
(cid:12) (cid:12) (cid:12) (cid:12)
(cid:12) (cid:12) (cid:12) (cid:12)
(cid:12) (cid:12) (cid:12) (cid:12)
using the definition of quasideterminan(cid:12)tsand the in(cid:12)var(cid:12)iancepropert(cid:12)ies describedin (4). Using the same
argument for n+i∈[0,n−1] but n+i 6=n−j−1 we see that Q(i,j)=0. Assuming n is arbitrarily
large we may summarise these properties of Q(i,j) as
−1 i+j+1=0
Q(i,j)= (25)
(0 (i<0 or j <0) and i+j+16=0.
Readers familiar with symmetric functions will recognise this property as analogous to a property of a
hook Schur function s (see [22, p47, Ex. 9]).
(i|j)
We shall call this type of quasideterminant a quasiwronskian. In the last section(see (9)) we showed
by means of Darboux transformationsthat if v is any givensolution of ncKP and Θ an n-rowvector of
0
eigenfunctions of L and M given by (7)–(8) then
v =v −2Q(0,0), (26)
0
6
alsosatisfiesncKP.Forsimplicitywewillchoosethevacuumsolutionv =0butthischoiceofvacuumis
0
not essential to what follows; the direct verificationcan be made for arbitraryvacuum but the formulae
are rather more complicated.
If we relabel and rescale the independent variables so that x =x, x =y, x =−4t, Θ satisfies the
1 2 3
linear equations
Θ =Θ ,
x2 xx (27)
Θ =Θ .
x3 xxx
WemayalsoallowΘtodependonhighervariablesxk andimposethenaturaldependenceΘxk =Θx···x.
kcopies
Now for any m, using the linear equations for Θ, we have
| {z }
n
∂ Θ e Θ e Θ e
Q(i,j)= n−j + k n−j
∂xm (cid:12)Θ(n+i+m) 0 (cid:12) (cid:12)Θ(n+i) 0 (cid:12)(cid:12)Θ(k−1+m) 0 (cid:12)
(cid:12) b (cid:12) Xk=1(cid:12) b (cid:12)(cid:12) b (cid:12)
(cid:12)(cid:12) n−1(cid:12)(cid:12) (cid:12)(cid:12) (cid:12)(cid:12)(cid:12)(cid:12) (cid:12)(cid:12)
=Q(cid:12) (i+m,j)+ (cid:12)Q(i,k)Q(cid:12)(m−1−k,j(cid:12))(cid:12). (cid:12) (28)
k=0
X
Using the conditions (25) the above simplifies considerably and we obtain
m−1
∂
Q(i,j)=Q(i+m,j)−Q(i,j+m)+ Q(i,k)Q(m−k−1,j). (29)
∂x
m
k=0
X
In particular
∂
Q(i,j)=Q(i+1,j)−Q(i,j+1)+Q(i,0)Q(0,j),
∂x
∂
Q(i,j)=Q(i+2,j)−Q(i,j+2)+Q(i,1)Q(0,j)+Q(i,0)Q(1,j),
∂x
2
∂
Q(i,j)=Q(i+3,j)−Q(i,j+3)+Q(i,2)Q(0,j)+Q(i,1)Q(1,j)+Q(i,0)Q(2,j).
∂x
3
Notethatthesesimplifiedformulae(29)areonlyvalidforsufficientlylargen. Forsmallernweshould
use (28) directly.
4.2 Derivatives of quasigrammians
Let us define
Ω(Θ,P) P†(j)
R(i,j)=(−1)j ,
Θ(i) 0
(cid:12) (cid:12)
(cid:12) (cid:12)
and call this type of quasideterminant a quasigram(cid:12)mian. As we h(cid:12)ave seen in (19), solutions obtained
(cid:12) (cid:12)
by binary Darboux transformation are of the form v = v −2R(0,0). As we did in the case of the
0
quasiwronskian type of solutions we choose v = 0 for simplicity. Hence Θ satisfies the same linear
0
equations as before and P, the vector of adjoint eigenfunctions, satisfies
P =−P , P =P .
x2 xx x3 xxx
Notethatchoiceofthetrivialvacuumisinessentialanddirectverificationcanbecompletedforarbitrary
vacuum.
7
Using (21), derivatives with respect to the x can be calculated;
m
Ω P†(j) Ω P†(j+m)
∂ R(i,j)=(−1)j +(−1)m+j−1
xm Θ(i+m) 0 Θ(i) 0
(cid:12) (cid:12) (cid:12) (cid:12)
(cid:12) (cid:12) (cid:12) (cid:12)
(cid:12) (cid:12) (cid:12) (cid:12)
(cid:12) (cid:12) (cid:12) (cid:12)
m−1 (−1)j+kΩ P†(k) Ω P†(j)
+
Θ(i) 0 Θ(m−1−k) 0
k=0 (cid:12) (cid:12)(cid:12) (cid:12)
X (cid:12) (cid:12)(cid:12) (cid:12)
m−1(cid:12) (cid:12)(cid:12) (cid:12)
(cid:12) (cid:12)(cid:12) (cid:12)
=R(i+m,j)−R(i,j+m)+ R(i,k)R(m−k−1,j).
k=0
X
Notice here that this final form for a derivative of a quasigrammian corresponds precisely with the
formula for the quasiwronskian (see (29)). Thus calculations in the subsequent sections carried out for
the quasiwronskiansolutions will be equally valid for the quasigrammiansolutions.
4.3 The commutative case
In order to better understand the derivative formulae we obtained above, we will assume that all quan-
tities commute and hence reduce to the familiar case of the commutative KP equation. Using (3), we
have
Θ
.
.
(cid:12) . (cid:12)
(cid:12) (cid:12)
(cid:12)Θ(n−2)(cid:12) Ω P
(cid:12) (cid:12)
(cid:12) Θ(n) (cid:12) Θ 0
Q(0,0)=−(cid:12) (cid:12), R(0,0)= (cid:12) (cid:12).
(cid:12) (cid:12) (cid:12) Ω (cid:12)
(cid:12) Θ (cid:12) (cid:12) (cid:12)
(cid:12) (cid:12) (cid:12) (cid:12)
(cid:12) (cid:12) (cid:12) (cid:12)
It is then simple to show that u = −2Q((cid:12)(cid:12)0b,0(cid:12)(cid:12))x = 2(log|Θ|)xx and u =(cid:12)−(cid:12)2R(0,0)x = 2(log|Ω|)xx which
are the well-knownsolutionsof the standardKP solutionin wronskianandgrammianformrespectively.
b
5 The direct approach
Returning to the noncommutative case, we will show directly that
v =−2Q(0,0), or v =−2R(0,0), (30)
8
aresolutionsofthe ncKPequation. To carryoutthis directverificationwefirstcalculatethe derivatives
of v
v =−2Q(0,0) =−2[Q(1,0)−Q(0,1)+Q(0,0)Q(0,0)]
x x
v =−2Q(0,0) =−2[Q(2,0)−Q(0,2)+Q(0,0)Q(1,0)+Q(0,1)Q(0,0)]
y y
v =−2Q(0,0) =8[Q(3,0)−Q(0,3)+Q(0,0)Q(2,0)+Q(0,1)Q(1,0)+Q(0,2)Q(0,0)]
t t
v =−2[Q(0,2)−2Q(1,1)+Q(2,0)−2Q(0,0)Q(0,1)+Q(0,0)Q(1,0)−Q(0,1)Q(0,0)
xx
+2Q(1,0)Q(0,0)+2Q(0,0)Q(0,0)Q(0,0)]
v =−2[Q(0,4)−2Q(2,2)+Q(4,0)
yy
+Q(0,0)Q(3,0)+Q(0,1)Q(2,0)−Q(0,2)Q(1,0)−Q(0,3)Q(0,0)
−2Q(0,0)Q(1,2)−2Q(0,1)Q(0,2)+2Q(2,0)Q(1,0)+2Q(2,1)Q(0,0)
+2Q(0,0)Q(1,0)Q(1,0)+2Q(0,0)Q(1,1)Q(0,0)
+2Q(0,1)Q(0,0)Q(1,0)+2Q(0,1)Q(0,1)Q(0,0)]
v =8[Q(0,4)−Q(1,3)−Q(3,1)+Q(4,0)
xt
+Q(0,0)Q(3,0)−Q(0,3)Q(0,0)−Q(0,0)Q(2,1)−Q(0,2)Q(0,1)−Q(0,1)Q(1,1)
−Q(0,0)Q(0,3)+Q(1,0)Q(2,0)+Q(1,1)Q(1,0))+Q(1,2)Q(0,0))+Q(3,0)Q(0,0)
+Q(0,0)Q(0,0)Q(2,0)+Q(0,0)Q(0,1)Q(1,0)+Q(0,0)Q(0,2)Q(0,0)
+Q(0,0)Q(2,0)Q(0,0)+Q(0,1)Q(1,0)Q(0,0)+Q(0,2)Q(0,0)Q(0,0)]
and v , which is straightforward but tedious to work out. This sort of calculation can be readily
xxxx
carried out using any computer algebra package that understands, or can be made to understand, non-
commutative multiplication. Substituting these into the ncKP equation(1) allterms exactly cancel and
the solution is verified. As remarked above, the derivative formulae are the same whether we use the
quasiwronskian or the quasigrammian formulation and so the above calculation simultaneously verifies
both types of solution.
6 Comparison with the bilinear approach
The direct approachto the determinantal solutions of the commutative KP equation is well known and
can be found in many places in the literature (see [14, 15] for example). Here we will compare it to the
alternative direct approach we studied above. In Hirota’s direct method, one first makes the change of
variables
u=2(logτ)
xx
and then rewrites (2) in bilinear form using Hirota derivatives
(D +D +3D )τ ·τ =0, (31)
xt xxxx yy
where [15]
∂m∂n
DmDnτ ·τ := τ(x+a,y+b)τ(x−a,y−b) .
x y ∂am∂bm
(cid:18) (cid:19)(cid:12)a,b→0
(cid:12)
The next step is to take a possible solution suchas a wronskianor grammian(cid:12)determinant andcalculate
(cid:12)
the derivatives with respect to x, y and t. So, for example, for a wronskiansolution we would take, (see
[14] for an explanation of the notation)
τ = Θ ,
(cid:12) (cid:12)
and calculate the derivatives (cid:12) (cid:12)
(cid:12)b(cid:12)
τ =(0,··· ,n−2,n)=τ ,
x (1)
τ =(0,··· ,n−2,n+1)+(0,··· ,n−3,n−1,n)=τ +τ ,
xx (2) (12)
τ =τ −τ ,
y (2) (12)
9
andsoon. Hereweuseashorthandpartitionnotationwhichdenotestheextraderivativesaddedtoeach
row in the wronksian τ.
Substituting these into the left hand side of (31) we obtain a constant multiple of
τ τ −τ τ +τ τ . (32)
(22) (21) (1) (2) (12)
While it is initially not obvious that this expressionis identically zero, using the Laplace expansionof a
2n×2n determinant, one verifies that (32) is indeed zero and the verification is complete.
Forthenoncommutativecasetheapproachisquitesimilar,however,curiouslysomeofthestepstaken
inthebilinearapproacharenotneeded. First,wedonotneedaCole-Hopfstylechangeofvariables,since
the solution is expressed directly as a quasiwronskian. Second, once we have substituted the derivatives
into the nonlinear equation the resulting expression immediately vanishes without the need to consider
any quasideterminant identities.
Thefactthatnoidentitiesareneededisratherunexpectedbutacloserexaminationofthederivatives
ofQ(0,0)inthecommutativecaseisilluminating. Whenallquantitiescommutewemayuse(3)toobtain
τ
Q(i,j)=(−1)j−1 (i+1,1j),
τ
and in particular Q(0,0)=−τ /τ =−τ /τ. Calculating the t derivative of each side of this gives
(1) x
−1Q(0,0) =Q(3,0)−Q(0,3)+Q(0,0)Q(2,0)+Q(0,1)Q(1,0)+Q(0,2)Q(0,0)
4 t
τ +τ τ τ −τ τ +τ τ
(14) (4) (1) (3) (12) (2) (13) (1)
=− + ,
τ τ2
whereas
1 −τ 1 τ τ τ
x xt x t
− =− − +
4 τ 4 τ τ2
(cid:18) (cid:19)t (cid:16) (cid:17)
τ −τ +τ τ (τ −τ +τ )
(14) (22) (4) (1) (3) (21) (13)
=− + .
τ τ2
Note that the term τ cannot come from Q(i,j) for any i,j and that the two expression for the
(22)
derivatives only agree when one makes use of the identity τ τ −τ τ +τ τ = 0. So it seems
(22) (21) (1) (2) (12)
that, in some sense, the identity used by hand in verifying solutions in the bilinear approach, is used
automatically as derivatives are calculated in the quasideterminant approach.
7 Conclusions
InthispaperweconsideredtwotypesofquasideterminantsolutionsofthenoncommutativeKPequation.
As wellas showinghow they may be constructedby Darboux transformations,they turnout to be ideal
fordirectverificationofthe solutionandplaythesamerolethattheτ-function doesinthecommutative
case.
Therearesomeinterestingfeaturestothedirectapproachusingquasideterminants. First,itillustrates
that a bilinear form is not needed, and indeed we believe that it does not exist, in the noncommutative
case. The secondrather surprising feature is that, unlike the commutative case, no identity is needed to
complete the verification.
Noncommutative versions of other integrable equations we have studied, a noncommutative Hirota-
Miwa equation [16, 18] and modified KP equation [17], also have quasideterminant solutions. However,
in these cases, direct verificationdoes require the use of quasideterminant identities of the form (5) and
so it seems that the ncKP equation may be exceptional in this respect.
References
[1] M. Hamanaka and K. Toda, Phys. Lett. A 316 77–83 (2003).
[2] L. D. Paniak, “Exact Noncommutative KP and KdV Multi-solitons” hep-th/0105185(2001).
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