Darboux transformation and exact multisolitons of CPN nonlinear sigma model

J. Math. Anal. Appl. 447 (2017) 1080–1101

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Darboux transformation and exact multisolitons of CP N nonlinear sigma model U. Saleem ∗ , M. Hassan Department of Physics, University of the Punjab, Quaid-e-Azam Campus, Lahore 54590, Pakistan

a r t i c l e

i n f o

Article history: Received 30 November 2015 Available online 27 October 2016 Submitted by J. Lenells Keywords: CP N model Multisolitons Darboux transformation Quasideterminant

a b s t r a c t Exact multisolitons of CP N sigma model are obtained by using Darboux transformation in terms of quasideterminant Darboux matrix. The multisoliton solutions are shown to be expressed in terms of quasideterminants of Gelfand and Retakh (1991) [10]. The method of quasideterminant Darboux matrix has been shown to be related with the dressing method that has already been used to compute iteratively exact multisolitons of the CP N sigma model by means of nonlinear superposition principle of Bäcklund transformations. Explicit expressions of one, two and three soliton solutions of CP 1 model have been computed by using properties of quasideterminants. © 2016 Elsevier Inc. All rights reserved.

1. Introduction Two-dimensional CP N sigma model is an important example of integrable field theories. It belongs to a general class of two-dimensional integrable field theories known as symmetric space nonlinear sigma models. It is a nonlinear sigma model with target space CP N = SU (N + 1)/S[U (N ) × U (1)]. The nonlinear sigma models share many features of four dimensional nonabelian gauge theories such as conformal invariance, confinement, asymptotic freedom, existence of topological charge taking integer values, etc. The classical and quantum integrability of nonlinear sigma model has been investigated in many articles (see e.g. [4,14, 1–3,5,7,6,15,29,20,22,21,31,16]). More recently, the CP N sigma models have also been studied in the context of integrability in string theory on AdS spaces and in the construction of giant magnon solutions by using the well-known soliton solution generating techniques of integrable systems (see e.g. [23,32,24,25,12] and references therein). As a classical integrable field theory, the CP N model possesses Lax pair and exhibits existence of an infinite number of local as well as nonlocal conservation laws. The solution generating method of Bäcklund * Corresponding author. E-mail addresses: [email protected], [email protected] (U. Saleem), [email protected] (M. Hassan). http://dx.doi.org/10.1016/j.jmaa.2016.10.045 0022-247X/© 2016 Elsevier Inc. All rights reserved.

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transformation of the CP N model has been investigated in [29,20,22,21,31]. In the previous studies the solutions of the CP N model were generated by iteratively using Bäcklund transformations. The Riccati equations and the corresponding Bäcklund transformations were used to develop a nonlinear superposition principle resulting in dressing method of generating multisoliton solutions of the model. The purpose of this paper is to study the multisoliton solutions of the CP N model by using the Darboux transformation where it is expressed in terms of quasideterminant Darboux matrix and the solutions are expressed in terms of quasideterminants. The method of quasideterminant Darboux matrix has been shown to be related to the dressing method which has already been employed to generate multi-soliton solutions of the CP N sigma model. We also obtain explicit expressions of one, two and three soliton solutions of CP 1 model by using the properties of quasideterminants. The CP N sigma model involves maps from R1,1 to CP N i.e. z : R1,1 −→ CP N , x± =

1 (t ± x) −→ z = (z 1 , · · · , z N +1 ) ∈ CN +1 , 2

where the homogeneous coordinates z = (z 1 , · · · , z N +1 ) obey z → z  = λz for λ = 0. The CP N sigma model is a locally U (1) invariant field theory defined by the Lagrangian density1 L=

1 (D+ z)† .D− z, 4

(1.1)

where † denotes the hermitian conjugation and the projective invariance implies |z|2 = (z † .z) =

N +1 

†

z i z i = 1,

i=1

and D± are the U (1) covariant derivatives which act on z : R1,1 −→ CP N as D± z = ∂± z − zA± ,   where A± = z † .∂± z are the components of the U (1) gauge field associated with the U (1) gauge symmetry. The Euler–Lagrange equations of CP N model resulting from the Lagrangian are   D+ D− z + (D+ z)† . (D− z) z = 0.

(1.2)

For N = 1, we have the simplest sigma model, the CP 1 model which is equivalent to the O(3) sigma model. The equivalence is established by taking φi = z † σi z, where σi are Pauli spin matrices. In this case 1 1 (D+ z)† .D− z → ∂+ φi ∂− φi , 4 4 and |z|2 = 1 → φi φi = 1.

L=

(1.3)

For our purpose we work with a U (N ) valued field g = (z, Y ) where z ∈ CN +1 and Y is a complex N ×N +1 matrix such that g is a moving orthogonal frame of CN +1 . Now the Euler–Lagrange equations are expressed in terms of the U (1) × U (N ) gauge fields  ± = A

z † ∂± z 0

 0 Y † ∂± Y

,

1 The spacetime conventions are such that light-cone coordinates x± are related to the orthonormal coordinates by x± = with derivatives ∂± = 12 (∂t ± ∂x ).

(1.4)

1 2

(t ± x)

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 ± acts on the matrix field g as and the U (1) × U (N ) covariant derivative D ± .  ± g = ∂± g − g A D

(1.5)

The CP N field equations are readily obtained from the Lagrangian of the model  + gg −1 D  +D  −g − D  − g = 0. D

(1.6)

The Lax pair of reduced CP N model is written as [5,7] ∂+ ψ(λ) = (λA + C) ψ(λ), ∂− ψ(λ) = λ−1 Bψ(λ),

(1.7)

where ψ(λ) is N + 1 component vector (eigenvector), λ is the spectral parameter and A, B and C are (N + 1) × (N + 1) matrices defined by ⎛

0 ⎜ −1 ⎜ ⎜ 0 A=⎜ ⎜ ⎜ .. ⎝ . 0 ⎛

1 0 0 .. . 0

0 0 0 .. . 0

0 ⎜ −ϕ 2 ⎜ ⎜ ⎜ −ϕ 3 B=⎜ .. ⎜ ⎝ . −ϕN +1 ⎛ c1 0 ⎜ 0 −c 1 ⎜ ⎜ ⎜ 0 −¯ c 2 C=⎜ .. ⎜ .. ⎝ . . 0 −¯ cN

⎞ 0  0⎟ ⎟  ⎟ J O 0⎟ = , ⎟ O O .. ⎟ .⎠ 0 ⎞ ϕ¯3 · · · ϕ¯N +1  0 ··· 0 ⎟ ⎟  ⎟ B1 B2† 0 ··· 0 ⎟= , ⎟ −B2 O .. .. ⎟ .. . . . ⎠ 0 ··· 0 ⎞ ··· 0  · · · cN ⎟ ⎟  ⎟ C1 C2 ··· 0 ⎟= . ⎟ −C2† O .. ⎟ .. . . ⎠ ··· 0

··· ··· ··· .. . ··· ϕ¯2 0 0 .. . 0 0 c2 0 .. . 0

(1.8)

The fields ϕj and cj are complex fields with the following properties c¯1 + c1 = 0, N +1 

ϕ¯i ϕi = 1,

(1.9) (1.10)

i=2

and c1 , ck are expressed in terms of fields ϕi as N ϕN +1 ∂+ ϕ¯N +1 + k=2 ∂+ ϕk ϕ¯k , c1 = 3|ϕN +1 |2 − 1 ck =

∂+ ϕk ϕk + c1 . ϕN +1 ϕN +1

(1.11)

The compatibility condition (∂+ ∂− ψ = ∂− ∂+ ψ) of the Lax pair (1.7) leads to the following matrix equations

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∂+ B + [B, C] = 0, ∂− C + [A, B] = 0.

(1.12)

By using (1.8), the matrix equations (1.12) give the following matrix equations ∂+ B1 + [B1 , C1 ] + C2 B2 − B2† C2† = 0, ∂+ B2 − C2† B1 + B2 C1 = 0,

∂+ B2† + B1 C2 − C1 B2† = 0,

∂− C1 + [J, B1 ] = 0, ∂− C2 + JB2† = 0,

∂− C2† − B2 J = 0.

(1.13)

Again by using (1.8), the matrix equations (1.13) reduce to the following scalar field equations ∂+ ϕ2 + 2ϕ2 c1 + ϕ3 c2 + . . . + ϕN +1 cN = 0, ∂+ ϕk+1 − ϕk c¯k − ϕk c1 = 0,

∂− ck − ϕ¯k+1 = 0,

k = 2, 3, . . . N + 1,

∂− c1 − ϕ2 + ϕ¯2 = 0.

(1.14)

The model we have been considering is the Minkowski model, i.e. a harmonic map from two-dimensional Minkowski space to CP N . This model has classical solutions of soliton type and we shall call these as the soliton solutions of the CP N model. The other type of models are the Euclidean CP N models. The finite action solutions are instantons and anti-instantons. The solutions can be obtained by the so called holomorphic method. In this paper, we apply matrix Darboux transformation to obtain quasideterminant multisoliton solutions of the Minkowski CP N model.  1 It is also convenient to define the CP N sigma model field z as z = |ff | , where |f | = f † .f 2 . The Lagrangian of the model now takes the form [16]   L= 2

2

2

|∂+ f | + |∂− f |

 dx+ dx− ,

4

|f |

†

where |∂± f | = (∂± f ) . (∂± f ). The Euler–Lagrange equation for the field f is  1−

f ⊗ f†



2

|f |

 ∂+ ∂− f − ∂+ f

f † .∂− f 2

|f |



 − ∂− f

f † .∂+ f 2

|f |

 = 0.

The Euler–Lagrange equations can also be expressed as the compatibility condition of the following set of linear equations (Lax pair) 2 K+ ψ, 1+λ 2 K− ψ, ∂− ψ = 1−λ ∂+ ψ =

(1.15)

where the (N +1) ×(N +1) matrix fields K± are given by K± = [∂± P, P ], with P being an (N +1) ×(N +1) projector matrix defined by P =

f ⊗ f† 2

|f |

,

P † = P,

P 2 = P.

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The compatibility condition of the Lax pair (1.15) is the equation of motion ∂+ K− − ∂− K+ = 0.

(1.16)

The equation of motion (1.16) is also a conservation equation. 2. Darboux transformation In this section, we employ Darboux transformation to generate multisoliton solutions of the CP N sigma model. Darboux transformation is one of well-known techniques of generating multisoliton solutions of many integrable systems (for more detail on Darboux transformation see e.g. [28,17]). For the CP N model, we introduce an (N + 1) × (N + 1) Darboux matrix D(x+ , x− , λ). The Darboux matrix acts on the solution  The covariance of the Lax pair under the Darboux ψ to the Lax pair (1.7) to give another solution ψ. transformation then gives the required transformation on the solutions to the field equations of the CP N model. The procedure has been explained as follows. We define Darboux transformation on the matrix solution ψ to the Lax par (1.7) as  ψ(λ) = D(x+ , x− , λ)ψ(λ).

(2.1)

For the Lax pair (1.7) to be covariant under the Darboux transformation (2.1), we require    +C  ψ, ∂+ ψ = λA   ψ. ∂− ψ = λ−1 B

(2.2)

By substituting equation (2.1) in equations (2.2), we get the following conditions on the Darboux matrix D(λ)       ∂+ D(λ) = λ AD(λ) − D(λ)A + CD(λ) − D(λ)C ,    ∂− D(λ) = λ−1 BD(λ) − D(λ)B .

(2.3)

For our system, we make the following ansatz for the Darboux matrix D(x+ , x− , λ) = λI − M (x+ , x− ; λ),

(2.4)

where M (x+ , x− ; λ) is some (N + 1) × (N + 1) matrix to be determined and I is an (N + 1) × (N + 1) identity matrix. In order to determine the Darboux matrix D(x+ , x− , λ), it is only necessary to determine the matrix M (x+ , x− ; λ). Now substituting (2.1) in equation (2.3) and using (1.7), we get the following Darboux transformation for the matrices A, B and C  = A, A  = M BM −1 , B  = C + [A, M ], C

(2.5)

and the matrix M is subjected to satisfy the following conditions ∂+ M = [C, M ] + [A, M ]M,

(2.6)

∂− M = B − M BM −1 .

(2.7)

U. Saleem, M. Hassan / J. Math. Anal. Appl. 447 (2017) 1080–1101

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The next task is to determine the matrix M in terms of particular solutions to the Lax pair, so that the explicit Darboux transformation in terms of particular solutions of the Lax pair, can be constructed. Let λ1 , · · · , λN +1 be N + 1 distinct real (or complex) constant parameters. Let us also define N + 1 constant column vectors |1 , |2 , · · · , |N + 1, such that Θ = (ψ(λ1 ) |1 , · · · , ψ(λN +1 ) |N + 1) = (|θ1  , · · · , |θN +1 ) ,

(2.8)

be an invertible (N + 1) × (N + 1) matrix. Each column |θi  = ψ(λi ) |i in M is a column solution to the Lax pair (1.7) when λ = λi , i.e., it satisfies ∂+ |θi  = (λi A + C) |θi  , ∂− |θi  = λ−1 i B |θi  ,

(2.9) (2.10)

and i = 1, 2, . . . , N + 1. If we define an (N + 1) × (N + 1) matrix of particular eigenvalues as Λ = diag(λ1 , . . . , λN +1 ),

(2.11)

then the Lax pair (2.9)–(2.10) can be written in (N + 1) × (N + 1) matrix form as ∂+ Θ = AΘΛ + CΘ, ∂− Θ = BΘΛ

−1

,

(2.12) (2.13)

where the (N + 1) × (N + 1) matrix M is a particular matrix solution of the Lax pair (1.7) with Λ being a matrix of particular eigenvalues. In terms of particular matrix solution Θ of the Lax pair (1.7), we define the matrix M to be M = ΘΛΘ−1 .

(2.14)

Now we show that the matrix M defined in (2.14), satisfies equations (2.6)–(2.7). By taking the x+ derivatives of the matrix (2.14), we have ∂+ M = ∂+ ΘΛΘ−1 + ΘΛ∂+ Θ−1 = (AΘΛ + CΘ)ΛΘ−1 − ΘΛΘ−1 ∂+ ΘΘ−1 = AΘΛ2 Θ−1 + CΘΛΘ−1 − ΘΛΘ−1 AΘΛΘ−1 − ΘΛΘ−1 C = [C, M ] + [A, M ]M.

(2.15)

Similarly by taking the x− derivative of the matrix (2.14), we have ∂− M = ∂− ΘΛΘ−1 − ΘΛΘ−1 ∂− ΘΘ−1 = B − ΘΛΘ−1 BΘΛ−1 Θ−1 = B − M BM −1 .

(2.16)

This shows that the choice (2.14) of the matrix M satisfies all the conditions imposed by the covariance of the Lax pair under the Darboux transformation. Therefore, we say that the transformation ψ = (λI − ΘΛΘ−1 )ψ,  = A, A

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U. Saleem, M. Hassan / J. Math. Anal. Appl. 447 (2017) 1080–1101

 = (ΘΛΘ−1 )B(ΘΛΘ−1 )−1 , B  = C + [A, ΘΛΘ−1 ], C

(2.17)

is the required Darboux transformation of the CP N model, in terms of particular matrix solution Θ to the Lax pair (2.12)–(2.13) with the particular eigenvalue matrix Λ. One-fold Darboux transformation can be expressed in terms of quasideterminant. We will use the notion of quasideterminant introduced by Gelfand and Retakh [10,11,26] (for details see Appendix A.1). Quasideterminants have found various applications in the theory of integrable systems, where multisoliton solutions of various integrable systems are expressed in terms of quasideterminants (see e.g. [8,9,27,18,13,19,30]). By introducing the notation ψ = ψ[1], ψ (1) = λψ and Θ(1) = ΘΛ, one-fold Darboux transformation in terms of quasideterminants, is expressed as   ψ[1] = (λI − M ) ψ = λI − ΘΛΘ−1 ψ    Θ ψ   =  (1) . Θ ψ (1) 

(2.18)

Similarly A[1] = A,   Θ  B[1] =  (1) Θ 

  I I   Θ  B  (1) O  Θ O    Θ I   C[1] = C − A,  (1)  , Θ O 

−1    ,  (2.19)

where O is an (N + 1) × (N + 1) null matrix. The iteration of Darboux transformation K times gives quasideterminant matrix solution to the CP N sigma model. For each k = 1, 2, · · · , K, let Θk be an invertible matrix solution to the Lax pair (1.7) at Λ = Λk . Now using the notation Θ(k) = ΘΛk , Θ[0] = Θ1 , ψ[0] = ψ and for K ≥ 1 we write −1

ψ[K] = ψ (1) [K − 1] − Θ(1) [K − 1] (Θ[K − 1])    Θ1 ··· ΘK ψ     .. .. .. ..   . . . .   =  (K−1) . (K−1) Θ (K−1)  · · · ΘK ψ  1   (K)  (K) (K)   Θ ··· ΘK ψ 1

ψ[K − 1]

(2.20)

Similarly the expressions for A[K], B[K] and C[K] are A[K] = A,   Θ1   ..   .  B[K] =  Θ(K−2)  1(K−1) Θ  1  (K)  Θ1

··· .. .

ΘK .. . (K−2)

···

ΘK

···

ΘK

···

(K−1) (K)

ΘK

    Θ1     ..     .    (K−2)  B O   Θ1   (K−1) O   Θ1   (K) O   Θ1 I .. .

··· .. .

ΘK .. . (K−2)

···

ΘK

···

ΘK

···

(K−1) (K)

ΘK

−1       O  ,  O   O  I .. .

(2.21)

U. Saleem, M. Hassan / J. Math. Anal. Appl. 447 (2017) 1080–1101

  Θ 1  ⎢  .. ⎢  ⎢  . ⎢   (K−2) A, C[K] = C − ⎢ ⎢  Θ1 ⎢  (K−1) ⎢ Θ ⎣  1  (K)  Θ1 ⎡

··· .. .

ΘK .. .

O .. .

(K−2)

···

ΘK

···

ΘK

(K−1) (K)

···

⎤   ⎥ ⎥ ⎥ ⎥ ⎥ O ⎥ . ⎥ I ⎥ ⎦  O 

1087

ΘK

(2.22)

The above results can be proved by using mathematical induction and properties of quasideterminants. The inductive proof of expressions (2.20)–(2.21) is similar to the one presented in [27,18]. In what follows, we give a proof of the expression (2.22). The expression (2.22) is true for K = 1. We assume that the expression (2.22) is true for K = p and show that (2.22) is true for K = p + 1. For the inductive step, we proceed as follows. We start with C [p + 1] = C [p] + [A, Θp+1 [p]Λp+1 Θp+1 [p]] ⎡   Θ1 Θ2 ···  ⎢  Θ2 Λ2 ··· ⎢  Θ1 Λ1 ⎢  ⎢  . . .. .. .. = C − ⎢A,  . ⎢  ⎢  p−1 p−1 Θ2 Λ2 ··· ⎣  Θ1 Λ1  p  Θ1 Λ p Θ2 Λ ··· 1

  Θ 1  ⎢  ⎢  Θ1 Λ1 ⎢  ⎢  ..  +⎢ . ⎢A,  ⎢  ⎢  Θ1 Λp−1 1 ⎣   p  Θ1 Λ1 ⎡

  Θ1    Θ1 Λ1   .. ×  .   Θ1 Λp−1 1   p  Θ1 Λ1

Θp Λp .. . Θp Λp−1 p Θp Λpp

2

⎤ O  ⎥ O ⎥ ⎥ .. ⎥ . ⎥ ⎥ ⎥ I ⎦  O 

Θ2

···

Θp

Θp+1

Θ2 Λ2 .. .

··· .. .

Θp Λp .. .

Θp+1 Λp+1 .. .

Θ2 Λp−1 2

···

Θp Λp−1 p

Θp+1 Λp−1 p+1

Θ2 Λp2

···

Θp Λpp

Θp+1 Λpp+1

Θ2

···

Θp

Θp+1

Θ2 Λ2 .. .

··· .. .

Θp Λp .. .

Θp+1 Λp+1 .. .

Θ2 Λp−1 2

···

Θp Λp−1 p

Θp+1 Λp−1 p+1

Θ2 Λp2

···

Θp Λpp

Θp+1 Λpp+1

  Θ1 Λ1  ⎢  ⎢  Θ1 Λ21 ⎢  ⎢  .. = C − ⎢A,  . ⎢  ⎢  p ⎣  Θ1 Λ1   Θ1 Λp+1 ⎡

1

  Θ1 Λ 1  ⎢  ⎢  Θ1 Λ21 ⎢  ⎢  ..  +⎢ . ⎢A,  ⎢  ⎢  Θ1 Λp1 ⎣    Θ1 Λp+1 1 ⎡

Θp

Θ2 Λ2

···

Θp Λp

Θ2 Λ22 .. .

··· .. .

Θp Λ2p .. .

Θ2 Λp2

···

Θp Λpp

Θ2 Λp+1 2

···

Θp Λp+1 p

Θ2 Λ2

···

Θp Λp

Θ2 Λ22 .. .

··· .. .

Θp Λ2p .. .

Θ2 Λp2

···

Θp Λpp

Θ2 Λp+1 2

···

Θp Λp+1 p

        Λp+1      

−1 ⎤    ⎥  ⎥  ⎥  ⎥  ⎥  ⎥  ⎥  ⎥  ⎦  

⎤ O  ⎥ O ⎥ ⎥ .. ⎥ . ⎥ ⎥ ⎥ I ⎦  O   Θp+1 Λp+1   Θp+1 Λ2p+1    ..  .   p Θp+1 Λp+1    Θp+1 Λp+1 p+1 

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 −1 ⎤  Θ1  Θ2 ··· Θp Θp+1     ⎥  Θ1 Λ1 Θ2 Λ2 ··· Θp Λp Θp+1 Λp+1  ⎥   ⎥   ⎥ .. .. .. .. ..   ⎥ × . . . . .  ⎥  ⎥ p−1 p−1  p−1  Θ1 Λp−1 ⎥ Θ Λ · · · Θ Λ Θ Λ 2 2 p p p+1 p+1  1  ⎦   p p p p Θ2 Λ2 ··· Θp Λp Θp+1 Λp+1   Θ1 Λ1 ⎤ ⎡   Θ1 Λ1 Θ2 Λ2 ··· Θp Λp O   ⎢  ⎥ Θ2 Λ22 ··· Θp Λ2p O ⎥ ⎢  Θ1 Λ21 ⎢  ⎥ ⎢  .. .. .. .. ⎥ .. = C − ⎢A,  . . . . . ⎥ ⎥ ⎢  ⎢  ⎥ p Θ2 Λp2 ··· Θp Λpp I ⎦ ⎣  Θ1 Λ1    Θ1 Λp+1 Θ2 Λp+1 · · · Θp Λp+1 O  p 1 2  ⎡   Θ1 Λ 1 Θ2 Λ2 ··· Θp Λp Θp+1 Λp+1   ⎢   ⎢  Θ1 Λ21 Θ2 Λ22 ··· Θp Λ2p Θp+1 Λ2p+1  ⎢   ⎢   .. .. .. .. ..  ⎢  . + ⎢A,  . . . .  ⎢   ⎢  Θ1 Λp1 Θ2 Λp2 ··· Θp Λpp Θp+1 Λpp+1   ⎣   p+1 p+1  p+1  Θ1 Λp+1 Θ Λ · · · Θ Λ Θ Λ 2 2 p p p+1 p+1  1 ⎛ −1  Θ1 Θ2 ··· Θp O   ⎜  ⎜ Θ1 Λ1 Θ2 Λ2 ··· Θp Λ p O  ⎜  ⎜ .. .. .. ..  .. ×⎜   . . . . . ⎜  ⎜  p p p ⎜ Θ1 Λ Θ Λ · · · Θ Λ O  2 2 p p 1 ⎝   Θ1 Λp+1 Θ2 Λp+1 · · · Θp Λp+1 I  p 1 2  ⎞−1 ⎤  Θ1  Θ · · · Θ Θ 2 p p+1    ⎟ ⎥ Θ2 Λ2 · · · Θp Λp Θp+1 Λp+1 ⎟ ⎥  Θ1 Λ1  ⎟ ⎥  ⎟ ⎥ .. .. .. .. . ×  ⎟ ⎥ . . . .  ⎟ ⎥ ⎥  ⎟ Θ2 Λp2 · · · Θp Λpp Θp+1 Λpp+1 ⎠ ⎥  Θ1 Λp1 ⎦    Θ1 Λp+1 Θ2 Λp+1 · · · Θp Λp Θp+1 Λp+1  1 2 p+1 Now using noncommutative Jacobi identity (A.8) and homological relation (A.9), we arrive at   Θ1  ⎢  ⎢  Θ1 Λ1 ⎢  ⎢  Θ Λ2 ⎢  1 1  C[p + 1] = C − ⎢ .. ⎢A,  ⎢  . ⎢  ⎢  Θ Λp ⎣  1 1   Θ1 Λp+1 1 ⎡

Θ2

···

Θp+1

Θ2 Λ2

···

Θp+1 Λp+1

Θ2 Λ22 .. .

··· .. .

Θp+1 Λ2p+1 .. .

Θ2 Λp2

···

Θp Λpp+1

Θ2 Λp+1 2

···

Θp+1 Λp+1 p+1

⎤ O  ⎥ O ⎥ ⎥ O ⎥ ⎥ ⎥ .. ⎥ . . ⎥ ⎥ I ⎥ ⎦  O 

(2.23)

This completes the proof. The multisoliton solutions of CP N model have already been obtained in literature [2,3,29]. Here we have shown that multisoliton solutions of the model can be reformulated by using the nice properties of quasideterminants. The advantage of quasideterminant Darboux matrix is that the explicit expressions of

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soliton solutions can be concisely formulated in terms of quasideterminants and can be computed by simple algebraic operations. 3. Relation with dressing method In this section we relate quasideterminant Darboux matrix with dressing method of generating multisoliton solutions of the CP N sigma model. Multisolitons of the CP N sigma model have been constructed by using different techniques which include method of Bäcklund transformation, pseudopotential method and Riccati’s equations (for details see e.g. [29,20,22,21,31]). In the dressing method, analytical properties of quasideterminant Darboux matrix are studied in the complex λ-plane. In order to compare the previously applied techniques and the quasideterminants method, we proceed as follows. First of all, we move from matrix solutions to the Lax pair and the field equations. From equation (2.14), we have M Θ = ΘΛ.

(3.1)

Let |θi  and |θj  be the column solutions of the Lax pair (1.7) at λ = λi and λ = λj respectively i.e. M |θi  = λi |θi ,

i = 1, 2, . . . , p,

M |θj  = λj |θj ,

j = p + 1, p + 2, . . . , N + 1.

(3.2)

Now we take λi = μ and λj = μ ¯, we may write the matrix M as M = μP + μ ¯P ⊥ ,

(3.3)

where P is the hermitian projector (i.e. P † = P ) and satisfies P 2 = P and P ⊥ = I − P . The projector P is hermitian projection on a complex space and P ⊥ as projection on orthogonal space. Now equation (3.3) can also be written as M = (μ − μ ¯) P + μ ¯I,

(3.4)

where the hermitian projector can be expressed in terms of column solutions to the Lax pair as P =

|θi  θi | . θi |θi 

(3.5)

The equation (3.3) allows us to re-express the one-fold Darboux transformation (2.1) on the matrix solution Ψ in terms of hermitian projector P as:   μ−μ ¯ −1 P Ψ. (3.6) Ψ[1] ≡ (λ − μ) D(x, t; λ)Ψ = I − λ−μ ¯ The K-fold Darboux transformation (2.20) on the matrix solution Ψ can be obtained by repeated action of Darboux matrix and can be written as (take P [1] = P ) Ψ[K] =

K−1   l=0

 μK−l − μ ¯K−l I− P [K − l] Ψ. λ−μ ¯K−l

(3.7)

The K-fold Darboux transformation (2.21) on the matrix field B can be written as B[K] =

K−1 

I−

l=0

 K−1   μK−l − μ μ ¯m − μm ¯K−l I− P [K − l] B P [m] , 1−μ ¯K−l 1−μ ¯l m=1

(3.8)

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where the projector P [m] is given by P [m] =

|θi [m] θi [m]| θi [m]|θi [m]

(3.9)

along with the column solutions |θi [m] =

m−1 

 μm−l − μ ¯m−l I− P [m − l] |θi . λ−μ ¯m−l

l=0

(3.10)

The expressions (3.7) and (3.8) can also be written as sum of K terms

Ψ[K] =

K−1  k=0

 1 I− Rl Ψ, λ−μ ¯l

(3.11)

and B[K] =

K−1 

I−

l=0

 K−1 −1  1 1 I− Rl B Rm , 1−μ ¯l 1−μ ¯m m=0

(3.12)

where the matrix R is expressed in terms of column solutions to the Lax pair by the following expression

Rk =

N −1 

(k)

(μl − μ¯k )

l=0

(l)

|θi  θi | (k)

(l)

θi |θi 

.

(3.13)

The matrix R is referred to as the residue matrix and is related to a matrix called the soliton correlation matrix which obeys matrix Riccati equations. The matrix Riccati equations are linearized and are completely solved. An inductive procedure is then applied to explicitly solve an iterated sequence of such matrix Riccati equations resulting in nonlinear superposition formula for the Bäcklund transformations that gives rise to multisoliton solutions of the model. For details see e.g. [29,20,22,21,31]. 4. Multisoliton solutions of C P 1 sigma model In order to illustrate the construction of multisoliton solutions of CP N model, we take the example of CP 1 model (CP 1 sigma model is also known as the O(3) sigma model). Here we will derive explicit expressions of multi-soliton solutions in term of ratio of determinants from the quasideterminants expressions. The equations of motion for the O(3) model becomes ∂− c1 = ϕ2 − ϕ¯2 ,

∂+ ϕ¯2 = 2c1 ϕ¯2 ,

(4.1)

¯2 2 ∂+ ϕ where c1 = ϕ2|ϕ 2 . The equations (4.1) can also be written as the compatibility condition of the following 2| Lax pair [5,7]



 λ c1 ∂+ ψ = ψ, −λ −c1   0 ϕ¯2 −1 ∂− ψ = λ ψ, −ϕ2 0

(4.2)

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where the eigenvector ψ is expressed as 

 X Y

ψ=

.

˜ 1 , respectively. The Let X1 , Y1 and X1 , Y1 , be the scalar solutions to the Lax pair (4.2) at λ = λ1 and λ = λ 2 × 2 matrix Θ (2.8) can be expressed in terms of these particular solutions as 

X1 Y1

X1 Y1

Θ = (|θ1 , |θ2 ) =

 .

(4.3)

Different matrix solutions Θk to the Lax pair for eigenvalue matrices Λk can be written as  Θk = 

Xk Yk λk 0

Λk =

 Xk , Yk  0 ˜k . λ

(4.4)

˜ k = −λk , Xk = −Xk and Y2k = Yk , so that the matrices (4.4) become For the reduced system, we take λ  Θk =  Λk =

Xk Yk λk 0

 −Xk , Yk  0 . −λk

(4.5)

In the present case, we have  I2 =



 1 0 0 1

,

Θ1 =

X1 Y1

−X1 Y1



 ,

Λ1 =

 λ1 0

0 −λ1

 ,

Θ1 Λ1 =

 λ1 X1 λ1 Y 1

λ1 X1 −λ1 Y1

. (4.6)

The one-fold Darboux transformation on the scalar solutions X, Y to the Lax pair (4.2) and scalar fields ϕ2 and c1 which are solutions to the field equations (4.1), are given by      X  X −X1 1  −X1 0  1 1       X[1] =  Y1 Y1 0  X +  Y1 Y1 1  Y,      λ1 X1 λ1 X1 0   λ1 X1 λ1 X1 λ       X  X −X1 1  −X1 0   1  1     Y [1] =  Y1 Y1 0  X +  Y1 Y1 1  Y,      λ1 Y1 −λ1 Y1 λ   λ1 Y1 −λ1 Y1 0    ⎞ ⎛  X −X1 0   X1 −X1 1  1    ⎟ ⎜ c1 [1] = c1 − ⎝ Y1 Y1 1  +  Y1 Y1 0 ⎠ ,      λ1 X1 λ1 X1 0   λ1 Y1 −λ1 Y1 0  ⎞ ⎛  X −X1 1   1 ⎟ ⎜ ϕ2 [1] = ϕ2 − ∂− ⎝ Y1 Y1 0 ⎠ .    λ1 Y1 −λ1 Y1 0 

(4.7)

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It can be easily checked that   X 1    Y1   λ1 X1

−X1 Y1 λ1 X1

1 0 0

    X   1    =  Y1     λ1 Y1

−X1 Y1 −λ1 Y1

0 1 0

     = 0.  

The one-fold Darboux transformations (4.7) become (1)

X[1] = X (1) −

X1 Δ1 (X1 , Y1 , X, Y )[2] , Y = Y1 Δ2 (X1 , Y1 )[1] (1)

Y1 Δ2 (X1 , Y1 , X, Y )[2] , X= X1 Δ1 (X1 , Y1 )[1]     (1) (1) X1 Y1 Δ1 (X1 , Y1 )[1] c1 [1] = c1 + , + = −c1 + ∂+ log Y1 X1 Δ2 (X1 , Y1 )[1] Y [1] = Y (1) −

 ϕ2 [1] = −ϕ¯2

Δ2 (X1 , Y1 )[1] Δ1 (X1 , Y1 )[1]

2 ,

(4.8)

where X1 and Y1 are the particular scalar solutions to the Lax pair (4.2) at λ = λ1 and we have used the notation X (1) = λX, Y (1) = λY and the determinants Δ1 and Δ2 are defined by   Y  1 Δ1 (X1 , Y1 , X, Y )[2] =  (1)  X1

 Y  , X (1) 

  X  1 Δ2 (X1 , Y1 , X, Y )[2] =  (1)  Y1

Δ1 (X1 , Y1 )[1] = X1 ,

 X  , Y (1) 

Δ2 (X1 , Y1 )[1] = Y1 .

In order to obtain two-soliton solutions of the model, we proceed to apply Darboux transformation again. For the two-fold Darboux transformation, we take the matrices Θ1 , Θ2 , Λ1 and Λ2 to be  X1 Y1

Θ1 =  Λ1 =

λ1 0

 −X1 , Y1  0 , −λ1

 X2 Y2

Θ2 =  Λ2 =

λ2 0

 −X2 , Y2  0 . −λ2

(4.9)

 can be expressed as In terms of particular scalar solutions, X1 , Y1 and X2 , Y2 , the matrices Θ and Θ  Θ1 Θ1 Λ1

Θ=

 = Θ



Θ1 Λ21

⎛

⎞ X1 −X1 X2 −X2 ⎜ Y Y1 Y2 Y2 ⎟ Θ2 ⎜ 1 ⎟ =⎜ ⎟, ⎝ λ1 X1 λ1 X1 λ2 X2 λ2 X2 ⎠ Θ2 Λ2 λ1 Y1 −λ1 Y1 λ2 Y2 −λ2 Y2    2 2 2 2 X −λ X λ X −λ X λ 1 1 2 2 1 1 2 2 . Θ2 Λ22 = λ21 Y1 λ21 Y1 λ22 Y2 λ22 Y2 

(4.10)

The two-fold Darboux transformation on the X, Y solution to the Lax pair (4.2) and scalar fields ϕ2 and c1 are given by

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    X1  X1 −X1 X2 −X2 1  −X1 X2 −X2    Y  Y  Y Y Y 0 Y Y Y2 1 1 2 2 1 2  1      λ  X +  λ1 X1 λ1 X1 λ2 X2 λ2 X2 X[2] =  λ1 X1 λ1 X1 λ2 X2 λ2 X2     λ1 Y1 −λ1 Y1 λ2 Y2 −λ2 Y2  λ1 Y1 −λ1 Y1 λ2 Y2 −λ2 Y2 0      λ2 X1 −λ2 X1 λ2 X2 −λ2 X2  λ2 X1 −λ2 X1 λ2 X2 −λ2 X2 λ2  1 1 2 2 1 1 2 2     X1  X1 −X1 X2 −X2 0 −X1 X2 −X2 1     Y  Y  Y Y Y 1 Y Y Y 0 1 1 2 2 1 1 2 2       0 Y [2] =  λ1 X1 λ1 X1 λ2 X2 λ2 X2 λ  X +  λ1 X1 λ1 X1 λ2 X2 λ2 X2     λ1 Y1 −λ1 Y1 λ2 Y2 −λ2 Y2  λ1 Y1 −λ1 Y1 λ2 Y2 −λ2 Y2 0  λ    2 2 2 2 2 2 2 2   λ2 Y 1  λ Y λ Y λ Y 0 λ Y λ Y λ Y λ Y λ 1 1 1 2 2 2 2 1 1 1 1 2 2 2 2   ⎛  X1 −X1 X2 −X2 0   X1 −X1 X2 −X2  ⎜ Y Y1 Y2 Y2 0   Y1 Y1 Y2 Y2 ⎜ 1   ⎜ c1 [2] = c1 − ⎜ λ1 X1 λ1 X1 λ2 X2 λ2 X2 0  +  λ1 X1 λ1 X1 λ2 X2 λ2 X2   ⎜ ⎝ λ1 Y1 −λ1 Y1 λ2 Y2 −λ2 Y2 1   λ1 Y1 −λ1 Y1 λ2 Y2 −λ2 Y2     λ2 X1 −λ2 X1 λ2 X2 −λ2 X2 0   λ2 Y1 λ21 Y1 λ22 Y2 λ22 Y2 1 1 2 2 1 ⎞ ⎛  X1 −X1 X2 −X2 0   ⎜ Y Y1 Y2 Y2 0 ⎟ ⎟ ⎜ 1 ⎟ ⎜ ϕ2 [2] = ϕ2 − ∂− ⎜ λ1 X1 λ1 X1 λ2 X2 λ2 X2 1 ⎟ . ⎟ ⎜ ⎝ λ1 Y1 −λ1 Y1 λ2 Y2 −λ2 Y2 0 ⎠  2   λ Y1 λ21 Y1 λ22 Y2 λ22 Y2 0  1

 0  1   0  Y,  λ  0         Y,     0 0 1 0 0

⎞   ⎟ ⎟ ⎟ ⎟ , ⎟ ⎠  

(4.11)

Again it can be easily checked that   X1   Y  1   λ1 X1   λ1 Y 1  2  λ X1 1

−X1 Y1 λ 1 X1 −λ1 Y1 −λ21 X1

X2 Y2 λ 2 X2 λ2 Y 2 λ22 X2

−X2 Y2 λ2 X2 −λ2 Y2 −λ22 X2

0 0 1 0 0

    X1     Y   1    =  λ1 X1     λ1 Y 1   2   λ Y1 1

−X1 Y1 λ1 X1 −λ1 Y1 λ21 Y1

X2 Y2 λ 2 X2 λ2 Y 2 λ22 Y2

−X2 Y2 λ2 X2 −λ2 Y2 λ22 Y2

0 0 0 1 0

       = 0.    

By simplifying the expressions, the two-fold Darboux transformations (4.11) on the scalar solutions to the Lax pair and on the solutions to the field equations are given by  λ Y  1 1  ⎜ ⎜ 2  λ21 X1 ⎜ X[2] = ⎜λ +  ⎝  X1   λ1 Y 1 ⎛

⎞ ⎛  λ2 Y2   X1  ⎟ ⎜ 2 λ 2 X2  ⎟ ⎜  λ21 X1 ⎜ ⎟  ⎟ X + ⎜−λ   X ⎝ X2  ⎠  1    λ1 Y 1 λ2 Y 2 

Δ1 (X1 , Y1 , X2 , Y2 , X, Y )[3] , Δ2 (X1 , Y1 , X2 , Y2 )[2]  ⎞ ⎛ ⎛  Y Y2   1  2  ⎜ ⎜  λ1 Y1 λ22 Y2  ⎟ ⎜ ⎟ ⎜ ⎟ X + ⎜λ2 +   −λ Y [1] = ⎜ ⎜ ⎟ ⎜  ⎝ Y2  ⎠ ⎝  Y1    λ1 X1 λ2 X2 

⎞ X2  ⎟ λ22 X2  ⎟ ⎟ ⎟Y X2  ⎠  λ2 Y 2 

=

=

Δ2 (X1 , Y1 , X2 , Y2 , X, Y )[3] , Δ1 (X1 , Y1 , X2 , Y2 )[2]

 λ X  1 1  2  λ1 Y 1   Y  1   λ 1 X1

⎞ λ2 X2  ⎟ λ22 Y2  ⎟ ⎟ ⎟Y Y2  ⎠  λ2 X2 

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   ⎛   λ2 Y λ2 X  ⎞   1 1  λ21 X1 λ22 X2   2 2   ⎟ ⎜    X X Y Y ⎜ 1 2 1 2 ⎟ ⎜    ⎟ + c1 [2] = c1 + ⎜  ⎟ ⎝  λ1 Y1 λ2 Y2   λ1 X1 λ2 X2  ⎠      X1  Y1 X2  Y2    Δ1 (X1 , Y1 , X2 , Y2 )[2] = c1 + ∂+ log , Δ2 (X1 , Y1 , X2 , Y2 )[2] ⎞ ⎛    λ21 Y1 λ22 X2   ⎟ ⎜ Y2  ⎟ ⎜ Y1 ⎟ ϕ2 [2] = ϕ2 + ∂− ⎜ ⎟ ⎜  ⎝  λ1 X1 λ2 X2  ⎠    Y1 Y2   2 Δ2 (X1 , Y1 , X2 , Y2 )[2] = ϕ2 , Δ1 (X1 , Y1 , X2 , Y2 )[2] where the determinants Δ1 and Δ2 are defined by   X1 X2   (1) (1) Δ1 (X1 , Y1 , X2 , Y2 , X, Y )[3] =  Y1 Y2  (2) (2)  X1 X2   Y1 Y2   (1) (1) Δ2 (X1 , Y1 , X2 , Y2 , X, Y )[3] =  X1 X2  (2) (2)  Y1 Y2

 X   Y (1)  ,  X (2)   Y   X (1)  ,  Y (2) 

(4.12)

  X  1 Δ2 (X1 , Y1 , X2 , Y2 )[2] =  (1)  Y1

 X2  (1)  , Y2 

  Y  1 Δ1 (X1 , Y1 , X2 , Y2 )[2] =  (1)  X1

 Y2  (1)  . X2 

The expressions (4.12) give two-soliton solutions to the O(3) sigma model. These solutions of nonlinear field equations are expressed in terms of scalar solutions to the Lax pair (linear problem). The multi-soliton solutions can be obtained by iterating the Darboux transformation. It seems convenient to express K-fold Darboux transformation (2.20) in the following form ψ[K] = T (K) ψ,

(4.13)

where the 2 × 2 matrix T (K) is expressed in the following notation

T

(K)

  Θ1   ..  . =  (K−1)  Θ1   Θ(K) 1

··· .. . ··· ···

ΘK .. . (K−1)

ΘK

(K)

ΘK

       Θ =   λK−1 I2   Θ  λK I  I2 .. .

 ˜I(K)  λ , λK I2 

2

(K)  where ˜Iλ , Θ and Θ are 2K × 2, 2 × 2K and 2K × 2K matrices respectively and are given by

˜I(K) = λ  = Θ

 I2 

λI2

···

λK−1 I2

Θ2 ΛK ··· Θ1 ΛK 1 2 ⎛ Θ1 Θ2 ⎜ Θ Λ Θ2 Λ2 ⎜ 1 1 Θ=⎜ .. .. ⎜ ⎝ . . K−1 Θ1 ΛK−1 Θ Λ 2 2 1

T ,

ΘK ΛK K ··· ··· .. . ···

 ,

⎞ ΘK ΘK ΛK ⎟ ⎟ ⎟. .. ⎟ ⎠ . ΘK−1 ΛK−1 K

(4.14)

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(1) For one-fold Darboux transformation ˜Iλ = I2 . The matrix elements of the matrix T (K) in the expression (4.14) can be calculated as

(K)

Tij

(K) Tii

 Θ  = Θ   Θ  = Θ 

  j    ˜I(K)  ˜I(K)   Θ λ λ ,  =     K   λ I2 ij  Θ  0 i    i   ˜I(K)  ˜I(K)   Θ λ λ ,  =      λK I2 ii  Θ λK 

i, j = 1, 2,

i = j,

i = 1, 2.

(4.15)

i

   j  represents i-th row of Θ  and ˜I(K) represents j-th column of ˜I(K) . Here Θ λ λ i

Similarly, one can re-express expression for B[K] and C[K] as B[K] = B − ∂− S (K) ,  C[K] = C − A, S (K) .

(4.16) (4.17)

The 2 × 2 matrix S (K) is expressed in the following notation

S

(K)

  Θ1   ..   .  =  Θ(K−2)  1(K−1) Θ  1  (K)  Θ1

··· .. .

ΘK .. .

···

(K−2) ΘK (K−1) ΘK (K) ΘK

··· ···

         Θ O2  =  Θ    I2   O2  O2 .. .

 E(K)  , O2 

(4.18)

where 2 × 2K matrix E(K) is given by E(K) =

 O2

O2

···

T I2

.

For one-fold Darboux transformation E(1) = I2 . The components of the matrix S (K) in the expression (4.18) can be calculated as     (K) j    Θ E(K)    Θ E   (K) , i = j, i, j = 1, 2, Si,j =    =      Θ O  0 Θ   ij i     (K) i    Θ E(K)    Θ E   (K)  = 0, Si,i =   (4.19)  =      Θ O  0   Θ ii

i

     denotes the i-th row of the matrix Θ  and E(K) j is j-th column of E(K) . where Θ i

From equation (2.21) the K-fold Darboux transformations on the scalar fields ϕ2 and c1 are given by (K)

ϕ2 [K] = ϕ2 − ∂− S21 ,   (K) (K) c1 [K] = c1 − S1,2 + S2,1 .

(4.20) (4.21)

By iterating Darboux transformation K times to arrive at the K-soliton solutions to the integrable O(3) model, we get the following result

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X[K] =

Δ1 (Xk , Yk , X, Y )[K + 1] , Δ2 (Xk , Yk )[K]

Y [K] =

Δ2 (Xk , Yk , X, Y )[K + 1] . Δ1 (Xk , Yk )[K]

(4.22)

We also get the following expressions of the K-soliton solutions to the integrable system (4.1) 

Δ1 (Xk , Yk )[K] c1 [K] = (−1) c1 + ∂+ log Δ2 (Xk , Yk )[K] 2  Δ2 (Xk , Yk )[K] K , ϕ2 [K] = (−1) ω Δ1 (Xk , Yk )[K] K

 , (4.23)

where ω = ϕ2 (for K even) and ω = ϕ¯2 (for K odd) and the determinants are defined by the following expressions. For K odd, the determinants Δ1 and Δ2 are given by   Y1 ···   .. ..  . .  Δ1 (Xk , Yk , X, Y )[K + 1] =  (K−1) Y ···  1   X (K) ··· 1   X1 ···   . ..  .. .  Δ2 (Xk , Yk , X, Y )[K + 1] =   X (K−1) · · ·  1   Y (K) ··· 1

YK .. . (K−1)

YK

(K)

XK

XK .. . (K−1)

XK

(K)

YK

      , (K−1)  Y   (K)  X  X   ..  .  . (K−1)  X   Y (K)  Y .. .

(4.24)

For K even, we have the following determinants   X1 ···   . ..  .. .  Δ1 (Xk , Yk , X, Y )[K + 1] =   Y (K−1) · · ·  1   X (K) ··· 1   Y1 ···   .. ..  . .  Δ2 (Xk , Yk , X, Y )[N + 1] =  (K−1) X ···  1   Y (K) ··· 1

XK .. . (K−1)

YK

(K)

XK

YK .. . (K−1)

XK

(K)

YK

      , (K−1)  Y   X (K)    Y   ..  .  . (K−1)  X   (K)  Y X .. .

(4.25)

Kink solutions: Since the O(3) sigma model is equivalent to the sine-Gordon equation, therefore, one can relate the solutions of the O(3) sigma model with the kink solutions of the sine-Gordon equation. Here we show that the well-known kink solutions of the sine-Gordon equation can be obtained by using quasideterminant Darboux matrix. If we take ϕ2 = eiφ , c1 becomes c1 = 2i ∂+ φ, the equation of motion (4.1) reduces to the sine-Gordon equation ∂+ ∂− φ = sin φ, and the associated Lax pair (linear system) (4.2) becomes

(4.26)

U. Saleem, M. Hassan / J. Math. Anal. Appl. 447 (2017) 1080–1101

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Fig. 1. Evolution of the kink solution φ[1] given by Eq. (4.31) of the sine-Gordon equation for a1 = 1, b1 = i and λ1 = i.

 ∂+ ψ =



i 2 ∂+ φ

λ − 2i ∂+ φ

−λ 

∂− ψ = λ−1

0 −eiφ

e−iφ 0

ψ,  ψ.

(4.27)

In order to obtain explicit expressions of multikink solutions, we take φ = 0 (or ϕ2 = 1 and c1 = 0) as the seed (trivial) solutions of the sine-Gordon model (or O(3)-model) (4.26). The Lax pair (4.27) becomes ∂+ X = λY,

∂+ Y = −λX,

(4.28)

∂− X = λ−1 Y,

∂− Y = −λ−1 X.

(4.29)

The linear system of equations (4.28)–(4.29) is then integrated to give the following solution Xk (x+ , x− ; λ) = ak eηk + bk eηk , Yk (x+ , x− ; λ) = i (ak eηk − bk eηk ) ,

(4.30)

  − where ak , bk are constants of integration and ηk = i λk x+ + λ−1 . k x If we take a1 = 1 and b1 = i, the one-fold Darboux transformation on the solutions to the sine-Gordon model, now gives the following solutions  φ[1] = −2i log

a1 eη1 − b1 eη1 a1 eη1 + b1 eη1



  = π − 4 cot−1 e2η1 .

(4.31)

The solution (4.31) is referred as kink solution. The time evolution of the kink solution is plotted in Fig. 1. The two-fold Darboux transformation on the solutions to the sine-Gordon equation, now gives the twokink solution ⎞ ⎛  X2   X1 ⎜  (1) (1)  ⎟ Y2  ⎟ ⎜ Y1 ⎜ ⎟ (4.32) φ[2] = −2i log ⎜  ⎟ . Y Y ⎝ 1 2 ⎠  (1) (1)   X1 X2  The three-fold Darboux transformation on the solutions to the sine-Gordon equation, now gives the three-kink solution

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Fig. 2. If we take a1 = 1, b1 = i, a2 = −1, b2 = −i and λ1 = 2i and λ2 = −i, the two-kink solution φ[2] given by Eq. (4.31) of the of the of the sine-Gordon equation.

Fig. 3. If we take a1 = 1, a2 = 0.3, a3 = 4, b1 = i, b2 = 0.3i, b3 = 4i and λ1 = 0.4i, λ2 = 0.3i, λ3 = 1.9i, three-kink solution φ[3] given by Eq. (4.33) of the sine-Gordon equation.

⎛  Y (3)  ⎜  1(2) ⎜  X1 ⎜  (1) ⎜ Y ⎜ 1 φ[3] = −2i log ⎜  (3) ⎜ X ⎜ 1 ⎜  (2) ⎝  Y1  X (1) 1

(3)

Y2 (2) X2 (1) Y2 (3)

X2 (2) Y2 (1) X2

⎞ (3) Y3  (2)  ⎟ X3  ⎟ ⎟ (1) Y3  ⎟ ⎟ ⎟ . (3)  ⎟ X3  ⎟ (2)  ⎟ Y3  ⎠ (1)  X 

(4.33)

3

The time evolution of the two-kink and three-kink solutions (4.32)–(4.33) is plotted in Figs. 2 and 3 respectively. The multi-kink solutions obtained in this section for the sine-Gordon equation can also be constructed by other solution generating techniques e.g. Bäcklund transformation, prolongation structure, Hirota direct method, Riccati equation and dressing method. 5. Concluding remarks In this paper, we studied construction of multisoliton solutions to the CP N sigma model by using Darboux transformation via quasideterminant Darboux matrix. Multisoliton solutions of the CP N sigma model are expressed in terms of quasideterminants. We also showed that the quasideterminant Darboux matrix is related to the dressing operator which had already been used in the literature for generating solutions of the model. In order to obtain explicit expressions of one, two and three soliton solutions, we applied Darboux transformation and the properties of quasideterminants to the CP 1 sigma model and expressed

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1099

multisoliton solutions as ratios of determinants. By using the properties of quasideterminants, one can also attempt to compute explicit multisoliton solutions of CP 2 and CP 3 sigma models. We shall address these problems in a separate work. The present work can also be further extended to study noncommutative and supersymmetric generalizations of the CP N sigma model. Appendix A A.1. Quasideterminant Let us define quasideterminant introduced by Gelfand and Retakh [10,11,26]: Definition 1. Let R be an associative algebra (noncommutative in general). Let A = [aij ]n×n be a square matrix and Aij denote a non-singular matrix of size (n − 1) × (n − 1) obtained by deleting i-th row and j-th column of A. The quasideterminant |A|ij is defined by   Aij  |A|ij ≡   rij

cij aij

   −1  cij ,  = aij − rij Aij 

(A.1)

T

where rij = (ai1 , ai2 , . . . , a ˆij , . . . , ain ) and cij = (a1j , a2j , . . . , a ˆij , . . . , anj ) denote row and column vectors respectively. (Note: The element a ˆij removed from the rij and cij .) The matrix A has n2 quasideterminants defined by (A.1). Examples. For n = 1 we have one quasideterminant. For n = 2   a11 a12 A= , a21 a22 we have four different quasideterminants    a   11 a12  |A|11 =   = a11 − a12 a−1 22 a21 ,  a21 a22      a  11 a12  |A|21 =   = a21 − a22 a−1 12 a11 ,  a21 a22 

|A|12 |A|22

 a  11 =  a21  a  11 =  a21

(A.2)

a12 a22 a12 a22

    = a12 − a11 a−1 21 a22 ,      = a22 − a21 a−1 11 a12 . 

(A.3)

(A.4)

For   a  11   a21   a31

a12 a22 a32

    a13    |A11 |−1 |A11 |−1 a21  22 32 a23  = a11 − a12 a13  |A11 |−1 |A11 |−1 a31 23 33 a33   −1  −1  a   a   22 a23   22 a23  = a11 − a12   a21 − a12   a31  a32 a33   a32 a33   −1  −1 a  a   22 a23   22 a23  − a13   a21 − a13   a31 .  a32 a33   a32 a33 

Now consider a 2 × 2 block decomposition of an n × n matrix   A11 A12 A= . A21 A22

(A.5)

(A.6)

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The quasideterminant of n × n matrix expanded about the k × k matrix A22 is defined by |A|22

 A  11 ≡  A21

A12 A22

    = A22 − A21 A−1 11 A12 . 

(A.7)

The quasideterminants obey noncommutative Jacobi identity  A  11   A21   A31

A12 A22 A32

A13 A23 A33

    A   11 =   A31 

A13 A33

   A   11 −   A31

A12 A32

 A   11    A21

A12 A22

−1   A   11     A21

A13 A23

   . 

(A.8)

From the noncommutative Jacobi identity, we get the homological relation  A  11   A21   A31

A12 A22 A32

A13 A23 A33

   A   11    =  A21     A31

A12 A22 A32

O O I

 A   11    A21    A31

A12 A22 A32

A13 A23 A33

    .  

(A.9)

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