Lax pair representation and Darboux transformation of NC Painlevé-II equation

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by  Irfan Mahmood

LAX PAIR REPRESENTATION AND DARBOUX
TRANSFORMATION OF NC PAINLEVÉ-II EQUATION
M. IRFAN
Abstract. TheextensionofPainlevéequationstononcommutativespaces
hasbeenconsideringextensivelyinthetheoryofintegrablesystemsanditis
2 also interestingto explore some remarkableaspects of these equations such
1
asPainlevéproperty,Laxrepresentation,Darbouxtransformationandtheir
0
connection to well know integrable equations. This paper is devoted to the
2
Lax formulation, Darboux transformation and Quasideterminant solution
n
of noncommutative Painlevé second equation which is recently introduced
a
J by V. Retakh and V. Rubtsov.

4

] 1. Introduction
h
p The Painlevé equations were discovered by Painlevé and his colleagues when
-
h they were classifying the nonlinear second-order ordinary differential equations
t
a with respect to their solutions [1].The importance of Painlevé equations from
m mathematical point of view is because of their frequent appearance in the var-
[ ious areas of physical sciences including plasma physics, fiber optics, quantum

1 gravity and field theory, statistical mechanics, general relativity and nonlinear
v optics. The classical Painlevé equations are regarded as completely integrable
0
equations and obeyed the Painlevé test [2, 3, 4]. These equations are subjected
0
9 to the some properties such as linear representation, hierarchy, Darboux trans-
0
formation(DT) and Hamiltonian structure because of their reduction from in-
.
1
tegrable systems, i.e, Painlevé second (P-II) equation arises as reduction of
0
KdV equation [5, 6] .
2
1 The Noncommutative(NC) extension of Painlevé equations is quite interesting
:
v in order to explore their properties which they possess on ordinary spaces. NC
i
X spaces are characterized by the noncommutativity of the spatial co-ordinates,
r if xµ are the space co-ordinates then the noncommutativity is defined by
a [xµ,xν] = iθµ where parameter θµν is anti-symmetric tensor and Lorentz in-
⋆
variant and [xµ,xν] is commutator under the star product. NC field theories
⋆
on flat spaces are given by the replacement of ordinary products with the
Moyal-products and realized as deformed theories from the commutative ones.
Moyal product for ordinary fields f(x) and g(x) is explicitly defined by
i ∂ ∂
f(x)⋆g(x) = exp(2θµν∂x′µ∂x′′ν)f(x′)g(x′′)x=x′=x′′
i ∂f ∂g
= f(x)g(x)+ θµν +O(θ2).
2 ∂x′µ∂x′′ν
1
2 M. IRFAN
this product obeys associative property f ⋆ (g ⋆ h) = (f ⋆ g) ⋆ h, if we apply
the commutative limit θµν → 0 then above expression will reduce to ordinary
product as f ⋆g = f.g.
We are familiar that Lax equation is a nice representation of integrable sys-
tems, the form of Lax equation on deformed space is the same as it has on
ordinary space, here ordinary product is replaced by the star product. The
Lax equation involves two linear operators, these operators may be differen-
tial operators or matrices [7]-[12]. If A and B are the linear operators then
Lax equation is given by A = [B,A] where [B,A] is commutator under
t ⋆ ⋆
the star product, this Lax pair formulasim is also helpful to construct the
DT of integrable systems. Now consider a linear system Ψ = A(x,t)Ψ and
x
Ψ = B(x,t)Ψ, the compatibility of this system yields A −B = [B,A] which
t t x ⋆
is called zero curvature condition [13] -[16], further let we express the commu-
tator [,] and anti-commutator [,] without writing the ⋆ as subscript then
− +
zero curvature condition may be written as A −B = [B,A] .
t x −
In this paper, i have applied Lax pair formalism for the representation of
NC P-II equation and this work also involves the explicit description of DT of
this equation. Finaly, i derive the multi-soliton solution of NC P-II equation
in terms of quasideterminants.
1.1. Linear representation. Many integrablesystems possess the linear rep-
resentation on ordinary as well as on NC spaces, this representation is also
known by the Lax representation, matrix P-II equation on ordinary space has
this kind of representation [18]. Here, we will see that how the NC P-II equa-
tion arises from the compatibility condition of following linear systems
Ψ = A(z;λ)Ψ,Ψ = B(z;λ)Ψ,
λ z
where A(z;λ) and B(z;λ) are matrices. The compatibility condition Ψ =
zλ
Ψ implies
λz
A −B = [B,A] (1)
z λ −
aboveexpression issimilartozerocurvatureconditionandanalternativelinear
representation of NC P-II equation with matrices A and B defined as under
8iλ2 +iv2 −2iz −iv + 1Cλ−1 −4λv
A = z 4
(cid:18)iv + 1Cλ−1 −4λv −8iλ2 −iv2 +2iz (cid:19)
z 4
−2iλ v
B =
(cid:18) v 2iλ(cid:19)
where the λ is a commuting parameter and C is a constant. Now we can easily
evaluate the following values
iv v +ivv −iv −4λv
A −B = z z zz z
z λ (cid:18)ivzz −4λvz −ivzv −ivvz(cid:19)
LAX PAIR REPRESENTATION AND DARBOUX TRANSFORMATION OF NC PAINLEVÉ-II EQUATION3
and
a −b−4λv
BA−AB = z
(cid:18)b−4λvz −a (cid:19)
where
a = iv v +ivv
z z
b = 2iv3 −2i[z,v] +iC
+
now by using the above values in (1) we have the following expression
0 −iv +2iv3 −2i[z,v] +iC
zz + = 0
(cid:18)ivzz −2iv3 +2i[z,v]+ −iC 0 (cid:19)
and finally we get
v = 2v3 −2[z,v] +C (2)
zz +
equation (2) is similar to the NC P-II equation which is introduced by V.
Retakh amd V. Rubtsov [19].
2. NC Symmetric Functions and Lax Representation
This section consists the Lax representation of the set of three equations of
functions u , u and u , this set is equivalent to the NC P-II equation [19] .
0 1 1
Now consider the linear system
L ψ = λψ
t
the time evolution of ψ is given by
ψ = Pψ
t
and the above system is equivalent to the Lax equation
L = [P,L] (3)
t −
here λ is a spectral parameter and λ = 0. Now we take the Lax pair L,P in
t
the following form
L O O
1
L =  O L2 O 
O O L
3
 
and
P O O
1
P =  O P2 O 
O O P
3
 
where
1 0 1 0 −1 0
L = ,L = ,L =
1 (cid:18)−v0 −1(cid:19) 2 (cid:18)−v1 −1(cid:19) 3 (cid:18)−v2 1(cid:19)
and the elements of matrix P are given by
ρ 0 −ρ 0 −1 0
1 2
P = ,L = ,L =
1 (cid:18)0 −ρ1(cid:19) 2 (cid:18) 0 ρ2(cid:19) 3 (cid:18)12σ 1(cid:19)
4 M. IRFAN
where
1 1
ρ = −v − α v−1,ρ = −v + α v−1,σ = v −v +2v
1 2 0 0 2 2 1 1 0 1 2
2 2
and
0 0
O = .
(cid:18)0 0(cid:19)
When the above Lax matrices L and P are subjected to the Lax equation (3)
we get
′
v = v v +v v +α
0 2 0 0 2 0
′
v = −v v −v v +α
1 2 1 1 2 1
′
v = v −v
2 1 0
the above system can be reduced to NC P-II equation (2) by eliminating v
0
and v . For this Lax representation of symmetric functions , the quasidetermi-
1
nants |I +µL| can not be expressed in terms of the expansion of symmetrics
ii
functions [20].
3. A Brief Introduction of Quasideterminants
This section is devoted to a brief review of quasideterminants introduced by
GelfandandRetakh[21]. Quasideterminantsarethereplacement forthedeter-
minantformatriceswithnoncommutative entriesandthesedeterminantsplays
very important role to construct the multi-soliton solutions of NC integrable
systems [22, 23], by applying the Darboux transformation. Quasideterminants
are not just a noncommutative generalization of usual commuta- tive determi-
nants but rather related to inverse matrices, quasideterminants for the square
matrices are defined as if A = a be a n×n matrix and B = b be the inverse
ij ij
matrix of A. Here all matrix elements are supposed to belong to a NC ring
with an associative product. Quasideterminants of A are defined formally as
the inverse of the elements of B = A−1
|A| = b−1
ij ij
this expression under the limit θµν → 0 , means entries of A are commuting,
will reduce to
detA
|A| = (−1)i+j
ij
detAij
where Aij is the matrix obtained from A by eliminating the i-th row and the
j-th column. We can write down more explicit form of quasideterminants. In
order to see it, let us recall the following formula for a square matrix
A B −1 A−BD−1C)−1 −A−1B(D −CA−1B)−1
A = =
(cid:18)C D(cid:19) (cid:18)−(D −CA−1B)−1CA−1 (D −CA−1B)−1 (cid:19)
(4)
where A and D are square matrices, and all inverses are supposed to exist.
We note that any matrix can be decomposed as a 2 × 2 matrix by block
decomposition where the diagonal parts are square matrices, and the above
LAX PAIR REPRESENTATION AND DARBOUX TRANSFORMATION OF NC PAINLEVÉ-II EQUATION5
formula can be applied to the decomposed 2×2 matrix. So the explicit forms
of quasideterminants are given iteratively by the following formula
|A| = a −Σ a |Aij|−1a
ij ij p6=i,q6=j iq pq pj
the number of quasideterminant of a given matrix will be equal to the numbers
of its elements for example a matrix of order 3 has nine quasideterminants. It
is sometimes convenient to represent the quasi-determinant as follows
a ··· a ··· a
11 1j 1n
. . . . .
(cid:12) .. .. .. .. .. (cid:12)
(cid:12) (cid:12)
|A|ij = (cid:12)(cid:12)ai1 ··· aij ··· ain(cid:12)(cid:12).
(cid:12) . . . . . (cid:12)
(cid:12) .. .. .. .. .. (cid:12)
(cid:12) (cid:12)
(cid:12)a ··· a ··· a (cid:12)
(cid:12) in ni nn(cid:12)
(cid:12) (cid:12)
(cid:12) (cid:12)
Let us consider examples of matrices with order 2 and 3, for 2×2 matrix
a a
11 12
A =
(cid:18)a21 a22(cid:19)
now the quasideterminats of this matrix are given below
a a
|A| = 11 12 = a −a a−1a
11 (cid:12) a a (cid:12) 11 12 22 21
21 22
(cid:12) (cid:12)
(cid:12) (cid:12)
(cid:12) (cid:12)
a a
|A| = 11 12 = a −a a−1a
12 (cid:12)a a (cid:12) 12 22 21 12
21 22
(cid:12) (cid:12)
(cid:12) (cid:12)
(cid:12) (cid:12)
a a
|A| = 11 12 = a −a a−1a
21 (cid:12) a21 a22(cid:12) 21 11 12 22
(cid:12) (cid:12)
(cid:12) (cid:12)
(cid:12) (cid:12)
a a
|A| = 11 12 = a −a a−1a .
22 (cid:12)a21 a22 (cid:12) 22 21 11 12
(cid:12) (cid:12)
(cid:12) (cid:12)
Thenumber ofquasidetermin(cid:12)ant ofag(cid:12)ivenmatrixwillbeequaltothenumbers
of its elements for example a matrix of order 3 has nine quasideterminants.
Now we consider the example of 3×3 matrix, its first quasidetermints can be
evaluated in the following way
a a a
11 12 13
|A| = (cid:12) a a a (cid:12) = a −a Ma −a Ma −a Ma −a Ma
11 21 22 23 11 12 21 13 21 12 31 13 31
(cid:12) (cid:12)
(cid:12) a31 a32 a33(cid:12)
(cid:12) (cid:12)
(cid:12) a a −1(cid:12)
where M =(cid:12) 22 23 (cid:12), similarly we can evaluate the other eight quaside-
(cid:12) a a (cid:12)
32 33
(cid:12) (cid:12)
terminants o(cid:12)f this mat(cid:12)rix .
(cid:12) (cid:12)
6 M. IRFAN
4. Darboux transformation
Darboux transformations play a very important role to construct the multi-
soliton solutions of integrable systems these transformations are deduced from
the given linear system of integrable systems. To derive the DT of NC P-II
χ
equation we consider its linear systems with the column vector ψ = .
(cid:18)Φ(cid:19)
Now the linear system will become
χ 8iλ2 +iv2 −2iz −iv + 1Cλ−1 −4λv χ
= z 4 (5)
(cid:18)Φ(cid:19) (cid:18)iv + 1Cλ−1 −4λv −8iλ2 −iv2 +2iz (cid:19)(cid:18)Φ(cid:19)
λ z 4
χ −2iλ v χ
= . (6)
(cid:18)Φ(cid:19) (cid:18) v 2iλ(cid:19)(cid:18)Φ(cid:19)
z
The standard transformations [24, 25, 26] on χ and Φ are given below
χ → χ[1] = γΦ−γ Φ (γ )χ−1(γ )χ (7)
1 1 1 1 1
Φ → Φ[1] = γχ−γ χ (γ )Φ−1(γ )Φ (8)
1 1 1 1 1
where χ , Φ are arbitrary solutions at γ and χ (γ ) , Φ (γ ) are the particular
1 1 1 1
solutions at γ = γ of equations (5) and (6), these equations will take the
1
following forms under the transformations (7) and (8)
χ[1] 8iλ2 +iv2[1]−2iz −iv [1]+ 1Cλ−1 −4λv[1] χ[1]
= z 4
(cid:18)Φ[1](cid:19) (cid:18)iv [1]+ 1Cλ−1 −4λv[1] −8iλ2 −iv2[1]+2iz (cid:19)(cid:18)Φ[1](cid:19)
λ z 4
(9)
χ[1] −2iλ v[1] χ[1]
= . (10)
(cid:18)Φ[1](cid:19) (cid:18) v[1] 2iλ(cid:19)(cid:18)Φ[1](cid:19)
z
Now from (6) and equation (10) we have the following expressions
χ = −iλχ+vΦ (11)
z
Φ = iλΦ+vχ (12)
z
and
χ [1] = −iλχ[1]+v[1]Φ[1] (13)
z
Φ [1] = iλΦ[1]+v[1]χ[1]. (14)
z
Now substituting the transformed values χ[1] and Φ[1] in equation (13) and
then after using the (11) and (12) in resulting equation, we get
v[1] = Φ χ−1vΦ χ−1. (15)
1 1 1 1
Equation (15) represents the Darboux transformation of NC P-II equation,
where v[1] is a new solution of NC P-II equation, this shows that how the new
solution is related to the seed solution v. By applying the DT iteratively we
can construct the multi-soliton solution of NC P-II equation.
LAX PAIR REPRESENTATION AND DARBOUX TRANSFORMATION OF NC PAINLEVÉ-II EQUATION7
4.1. Quasideterminant solutions. The transformations (7) and (8) may be
expressed in the form of quasideterminants , first consider the equation (7) in
follows form
χ[1] = γ Φ −γ Φ (γ )χ−1(γ )χ
0 0 1 1 1 1 1 0
or
χ χ
χ[1] = 1 0 = δe[1] (16)
(cid:12)γ1Φ1 γ0Φ0 (cid:12) χ
(cid:12) (cid:12)
(cid:12) (cid:12)
similarly we can do for the equa(cid:12)tion (8) (cid:12)
Φ Φ
Φ[1] = 1 0 = δe[1] (17)
(cid:12)γ1χ1 γ0χ0 (cid:12) Φ
(cid:12) (cid:12)
(cid:12) (cid:12)
we have taken γ = γ , χ = χ a(cid:12)nd Φ = Φ in(cid:12)order to generalize the transfor-
0 0 0
mations in Nth form. Further, we can represent the transformations χ[2] and
Φ[2] by quasideterminants
χ χ χ
2 1 0
χ[2] = (cid:12)γ2Φ2 γ1Φ1 γ0Φ0 (cid:12) = δo[2]
(cid:12) (cid:12) χ
(cid:12)γ2χ γ2χ γ2χ (cid:12)
(cid:12) 2 2 1 1 0 0 (cid:12)
(cid:12) (cid:12)
(cid:12) (cid:12)
and
Φ Φ Φ
2 1 0
Φ[2] = (cid:12)γ2χ2 γ1χ1 γ0χ0 (cid:12) = δo[2].
(cid:12) (cid:12) Φ
(cid:12)γ2Φ γ2Φ γ2Φ (cid:12)
(cid:12) 2 2 1 1 0 0 (cid:12)
(cid:12) (cid:12)
here superscripts e and o of(cid:12)δ represent the even(cid:12)and odd order quasidetermi-
nants. The Nth transformations for δo[N] and δo[N] in terms of quasideter-
χ Φ
minants are given below
χ χ ··· χ χ
N N−1 1 0
(cid:12) γNΦN γN−1ΦN−1 ··· γ1χ1 γ0Φ0 (cid:12)
(cid:12) . . . . (cid:12)
δo[N] = (cid:12) .. .. ··· .. .. (cid:12)
χ (cid:12) (cid:12)
(cid:12)γN−1Φ γN−1Φ ··· γN−1χ γN−1Φ (cid:12)
(cid:12) N N N−1 N−1 1 1 0 0(cid:12)
(cid:12) (cid:12)
(cid:12)(cid:12) γNNχN γNN−1χN−1 ··· γ1Nχ1 γ0Nχ0 (cid:12)(cid:12)
(cid:12) (cid:12)
(cid:12) (cid:12)
and
Φ Φ ··· Φ Φ
N N−1 1 0
γ χ γ χ ··· γ χ γ χ
(cid:12) N N N−1 N−1 1 1 0 0 (cid:12)
(cid:12) . . . . (cid:12)
δo[N] = (cid:12) .. .. ··· .. .. (cid:12)
Φ (cid:12) (cid:12)
(cid:12)γN−1χ γN−1χ ··· γN−1χ γN−1χ (cid:12)
(cid:12) N N N−1 N−1 1 1 0 0(cid:12)
(cid:12) (cid:12)
(cid:12)(cid:12) γNNΦN γNN−1ΦN−1 ··· γ1NΦ1 γ0NΦ0 (cid:12)(cid:12)
(cid:12) (cid:12)
(cid:12) (cid:12)
8 M. IRFAN
here N is to be taken as even. in the same way we can write Nth quasideter-
minant representations of δe[N] and δe[N].
χ Φ
χ χ ··· χ χ
N N−1 1 0
γ Φ γ Φ ··· γ χ γ Φ
(cid:12) N N N−1 N−1 1 1 0 0 (cid:12)
(cid:12) . . . . (cid:12)
δe[N] = (cid:12) .. .. ··· .. .. (cid:12)
χ (cid:12) (cid:12)
(cid:12)γN−1χ γN−1χ ··· γN−1χ γN−1χ (cid:12)
(cid:12) N N N−1 N−1 1 1 0 0(cid:12)
(cid:12) (cid:12)
(cid:12)(cid:12) γNNΦN γNN−1ΦN−1 ··· γ1NΦ1 γ0NΦ0 (cid:12)(cid:12)
(cid:12) (cid:12)
and (cid:12) (cid:12)
Φ Φ ··· Φ Φ
N N−1 1 0
γ χ γ χ ··· γ χ γ χ
(cid:12) N N N−1 N−1 1 1 0 0 (cid:12)
(cid:12) . . . . (cid:12)
δe[N] = (cid:12) .. .. ··· .. .. (cid:12).
Φ (cid:12) (cid:12)
(cid:12)γN−1Φ γN−1Φ ··· γN−1Φ γN−1Φ (cid:12)
(cid:12) N N N−1 N−1 1 1 0 0(cid:12)
(cid:12) (cid:12)
(cid:12)(cid:12) γNNχN γNN−1χN−1 ··· γ1Nχ1 γ0Nχ0 (cid:12)(cid:12)
(cid:12) (cid:12)
Similarly, we can derive the expression for Nth soliton solution from equation
(cid:12) (cid:12)
(15) by applying the Darboux transformation iteratively, now consider
v[1] = Λφ[1]Λχ[1]−1vΛφ[1]Λχ[1]−1
1 1 1 1
where
Λφ[1] = Φ
1 1
Λχ[1] = χ
1 1
thisis onefoldDarbouxtransformation. Thetwo foldDarbouxtransformation
is given by
v[2] = φ[1]χ−1[1]v[1]φ[1]χ−1[1]. (18)
We may rewrite the equation (16) and equation (17) in the following forms
χ χ
χ[1] = 1 0 = Λχ[2]
(cid:12)γ1Φ1 γ0Φ0 (cid:12) 2
(cid:12) (cid:12)
(cid:12) (cid:12)
(cid:12) (cid:12)
Φ Φ
Φ[1] = 1 0 = Λφ[2].
(cid:12)γ1χ1 γ0χ0 (cid:12) 2
(cid:12) (cid:12)
(cid:12) (cid:12)
and equation (18) may be written as
(cid:12) (cid:12)
v[2] = Λφ[2]Λχ[2]−1Λφ[1]Λχ[1]−1vΛφ[1]Λχ[1]−1Λφ[2]Λχ[2]−1
2 2 1 1 1 1 2 2
In the same way, we can derive the expression for three fold Darboux trans-
formtion
v[3] = Λφ[3]Λχ[3]−1Λφ[2]Λχ[2]−1Λφ[1]Λχ[1]−1vΛφ[1]Λχ[1]−1Λφ[2]Λχ[2]−1Λφ[3]Λχ[3]−1.
3 3 2 2 1 1 1 1 2 2 3 3
here
χ χ χ
2 1 0
χ[2] = (cid:12)γ2Φ2 γ1Φ1 γ0Φ0 (cid:12) = Λχ[3]
3
(cid:12) (cid:12)
(cid:12)γ2χ γ2χ γ2χ (cid:12)
(cid:12) 2 2 1 1 0 0 (cid:12)
(cid:12) (cid:12)
(cid:12) (cid:12)
LAX PAIR REPRESENTATION AND DARBOUX TRANSFORMATION OF NC PAINLEVÉ-II EQUATION9
and
Φ Φ Φ
2 1 0
Φ[2] = (cid:12)γ2χ2 γ1χ1 γ0χ0 (cid:12) = Λφ[3].
3
(cid:12) (cid:12)
(cid:12)γ2Φ γ2Φ γ2Φ (cid:12)
(cid:12) 2 2 1 1 0 0 (cid:12)
(cid:12) (cid:12)
Finaly,byapplyingthetra(cid:12)nsformtioniterative(cid:12)lywecanconstructtheN-fold
Darboux transformation
v[N] = Λφ [N]Λχ[N]−1Λφ [N−1]Λχ [N−1]−1...Λφ[2]Λχ[2]−1Λφ[1]Λχ[1]−1vΛφ[1]Λχ[1]−1
N N N−1 N−1 2 2 1 1 1 1
Λφ[2]Λχ[2]−1...Λφ [N −1]Λχ [N −1]−1Λφ [N]Λχ[N]−1
2 2 N−1 N−1 N N
by considering the following substitution
Θ [N] = Λφ [N]Λχ[N]−1
N N N
in above expression, we get
v[N] = Θ [N]Θ [N −1]...Θ [2]Θ [1]vΘ [1]Θ [2]...Θ [N −1]Θ [N]
N N−1 2 1 1 2 N−1 N
or
v[N] = ΠN−1Θ [N −k]VΠ0 Θ [N −j]
k=0 N−k j=N−1 N−j
here we present only the Nth expression for odd order quasideterminants
Λφ [N] and Λχ[N]
N N
Φ Φ ··· Φ Φ
2N 2N−1 1 0
γ χ γ χ ··· γ χ γ χ
(cid:12) 2N 2N 2N−1 2N−1 1 1 0 0 (cid:12)
(cid:12) . . . . (cid:12)
Λφ [2N +1] = (cid:12) .. .. ··· .. .. (cid:12)
2N+1 (cid:12) (cid:12)
(cid:12)γ2N−1χ γ2N−1χ ··· γ2N−1χ γ2N−1χ (cid:12)
(cid:12) 2N 2N 2N−1 2N−1 1 1 0 0(cid:12)
(cid:12)(cid:12)(cid:12) γ22NNΦ2N γ22NN−1Φ2N−1 ··· γ12NΦ1 γ02NΦ0 (cid:12)(cid:12)(cid:12)
(cid:12) (cid:12)
and (cid:12) (cid:12)
χ χ ··· χ χ
2N 2N−1 1 0
γ Φ γ Φ ··· γ χ γ Φ
(cid:12) 2N 2N 2N−1 2N−1 1 1 0 0 (cid:12)
(cid:12) . . . . (cid:12)
Λχ [2N +1] = (cid:12) .. .. ··· .. .. (cid:12)
2N+1 (cid:12) (cid:12)
(cid:12)γ2N−1Φ γ2N−1Φ ··· γ2N−1χ γ2N−1Φ (cid:12)
(cid:12) 2N 2N 2N−1 2N−1 1 1 0 0(cid:12)
(cid:12)(cid:12)(cid:12) γ22NNχ2N γ22NN−1χ2N−1 ··· γ12Nχ1 γ02Nχ0 (cid:12)(cid:12)(cid:12)
(cid:12) (cid:12)
(cid:12) (cid:12)
5. Conclusion
In this paper, I focused on the linear representations of NC P-II equation
, I have also constructed its Darboux transformation. Finally I derived its
multi-soliton solution in terms of quasideterminants. The further motivations
are to explore its other aspects such that its connection to other integrable
equations, its hierarchy and Painlevé property.
10 M. IRFAN
6. Aknowledgement
I would like to thank V. Roubtsov, V. Retakh and M. Hassan for their
valuablediscussionstomeduringmyresearchworkonthispaperandespecially
to M. Cafasso who patiently explained me the matrix version of Lax pair for
NC Painlevé II from his paper [27] with M. Bertola. My special thanks to
the university of the Punjab, Pakistan, on funding me for my Ph.D project in
France. I am also thankful to S. Meljanac and Z. ˇSkoda for their valuable
suggestions andto thefranco-croatiancooperationprogramEgidePHCCogito
24829NH for a financial support of my visit to Zagreb University.
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