arXiv:1211.1762v1 [nlin.SI] 8 Nov 20123 Standard binary Darboux transformation To define the binary transformation we follow the approach of [13,40,41,42,43,45] and consider a space

Symmetry, Integrability and Geometry: Methods and Applications SIGMA 8 (2012), 084, 15 pages

Quasi-Grammian Solutions of the Generalized

Coupled Dispersionless Integrable System

Bushra HAIDER and Mahmood-ul HASSAN

Department of Physics, University of the Punjab,Quaid-e-Azam Campus, Lahore-54590, Pakistan

E-mail: [email protected], [email protected]

URL: http://pu.edu.pk/faculty/description/526/,http://www.pu.edu.pk/faculty/description/538/

Received June 22, 2012, in final form October 10, 2012; Published online November 08, 2012

http://dx.doi.org/10.3842/SIGMA.2012.084

Abstract. The standard binary Darboux transformation is investigated and is used toobtain quasi-Grammian multisoliton solutions of the generalized coupled dispersionless in-tegrable system.

Key words: integrable systems; binary Darboux transformation; quasideterminants

2010 Mathematics Subject Classification: 70H06; 22E99

1 Introduction

The interest in dispersionless integrable systems is due to their wide range of applicability invarious fields of mathematics and physics [1, 2, 8, 16, 17, 19, 20, 21, 22, 23, 24, 25, 26, 27, 47,48, 50]. Most of the dispersionless integrable systems belong to a family where these systemsarise as quasi-classical limit of ordinary integrable systems with a dispersion term [1, 2, 8, 22,23, 25, 27, 47, 48, 50]. But there are important examples of dispersionless integrable systemwhich are referred to as dispersionless not in the sense mentioned above but due to the absenceof dispersion term. The coupled dispersionless integrable systems and its generalizations areexamples of such integrable systems [16, 17, 19, 20, 21, 24, 26]. The Darboux transformation ofthe generalized coupled dispersionless integrable system has been studied in a recent work [16].

The purpose of this paper is to study the standard binary Darboux transformation of thegeneralized coupled dispersionless integrable system and to derive exact solutions in terms ofquasi-Grammians. We employ the method introduced in [15], construct standard binary Dar-boux transformation by introducing Darboux matrices of the system for the direct and theadjoint Lax pairs and then obtain binary Darboux matrix by composing the two Darboux trans-formations. We obtain the quasi-Grammian multisoliton solutions using the iterated binaryDarboux transformations. We also consider the system based on Lie group SU(N) and obtainexplicit solutions of the system based on SU(2).

The action of the generalized coupled dispersionless integrable system based on some non-Abelian Lie group G is given by

I =

∫dtdxL(S, Sx, St), (1.1)

where the Lagrangian density L(S, Sx, St) is defined by

L = Tr(

1

2SxSt −

1

3G[S, [Sx, S]]

),

arX

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211.

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mailto:[email protected]:[email protected]://pu.edu.pk/faculty/description/526/http://www.pu.edu.pk/faculty/description/538/http://dx.doi.org/10.3842/SIGMA.2012.084

2 B. Haider and M. Hassan

where S is a matrix field and G is a constant matrix taking values in the non-Abelian Liealgebra g of the Lie group G. The matrix fields S and G are Lie algebra g valued, i.e., S = φaT aand G = κaT a, where anti-hermitian generators {T a, a = 1, 2, . . . ,dim g} of the Lie algebra gobey [T a, T b] = fabcT c and Tr(T aT b) = −δab. For any X ∈ g, X = XaT a. Note that φa =φa (x, t) is a vector field with components {φq, a = 1, 2, . . . ,dim g} and κ is the constant vectorhaving components {κa, a = 1, 2, . . . ,dim g}. The equation of motion of the generalized coupleddispersionless system as obtained from (1.1) is

Sxt − [[S,G], Sx] = 0. (1.2)

For G = SU(2) we get from (1.2)

qxt + (rr̄)x = 0, rxt − 2qxr = 0, r̄xt − 2qxr̄ = 0, (1.3)

where q is a real valued function and r is a complex valued function of x and t. Here r̄ denotescomplex conjugate of r.

The generalized coupled dispersionless system (1.2) can be written as the compatibility con-dition of the following Lax pair

∂xψ = U(x, t, λ)ψ, ∂tψ = V (x, t, λ)ψ, (1.4)

where ψ ∈ G and λ is a real (or complex) parameter. The fields U and V are n×n matrix fieldsand are given by

U(x, t, λ) = λ∂xS, V (x, t, λ) = [S,G] + λ−1G.

The compatibility condition of the linear system (1.4) is the zero curvature condition

∂tU(x, t, λ)− ∂xV (x, t, λ) + [U(x, t, λ), V (x, t, λ)] = 0. (1.5)

Note that the above equation (1.5) is equivalent to the equation of motion (1.2). The Darbouxtransformation of the generalized coupled dispersionless system has been discussed in [16]. Inthe next section we will retrace the steps for the Darboux transformation for direct and adjointspaces and then we will combine the two elementary Darboux transformations to obtain thestandard binary Darboux transformation of the generalized coupled dispersionless system.

2 Darboux transformation on the direct and adjoint Lax pairs

In this section we discuss the Darboux transformation on the solutions to the direct and adjointLax pairs. For details of Darboux transformation see e.g. [3, 4, 5, 6, 7, 14, 18, 29, 30, 31, 32, 34,35, 36, 37, 38, 39, 44, 46, 49]. The one-fold Darboux transformation on the matrix solution tothe Lax pair (1.4) is defined by

ψ̃(λ) = D(x+, x−, λ)ψ(λ), (2.1)

where D(x, t, λ) is the Darboux matrix. We use the following ansatz for the Darboux matrixD(x, t, λ)

D(x+, x−, λ) = λ−1I −M(x+, x−), (2.2)

and M(x+, x−) is some n× n matrix field to be determined and I is an n× n identity matrix.The Darboux matrix transforms the matrix solution ψ in space V to a new solution ψ̃ in Ṽ, i.e.

D(λ) : V → Ṽ. (2.3)

Quasi-Grammian Solutions of the Generalized Coupled Dispersionless Integrable System 3

The new solution ψ̃ satisfies the Darboux transformed Lax pair

∂xψ̃ = Ũ(x, t, λ)ψ̃, ∂tψ̃ = Ṽ (x, t, λ)ψ̃, (2.4)

where the matrix-valued fields Ũ and Ṽ are given as

Ũ(x, t, λ) = λ∂xS̃, Ṽ (x, t, λ) =[S̃, G̃

]+ λ−1G̃,

and S̃ and G̃ are the Lie algebra valued transformed matrix fields. The covariance of theLax pair (1.4) under Darboux transformation can be checked by substituting equation (2.1) inequations (2.4). The covariance implies the following Darboux transformation on the matrix-valued fields S and G

S̃ = S −M, (2.5)G̃ = G. (2.6)

As mentioned earlier equation (2.6) shows that G is a constant matrix and the matrix M issubjected to satisfy the following equations

∂xMM = [∂xS,M ], ∂tM = [[S,G],M ] + [G,M ]M.

The matrix M can be written in terms of the solutions of the linear system [16]

M = ΘΛ−1Θ−1, (2.7)

where Θ is the particular matrix solution of the Lax pair defined by

Θ = (ψ(λ1)|1〉, . . . , ψ(λn)|n〉) = (|θ1〉, . . . , |θn〉),

Each column |θi〉 = ψ(λi)|i〉 in Θ is a column solution of the Lax pair (1.4) when λ = λi, i.e., itsatisfies

∂x|θi〉 = λi∂xS|θi〉, ∂t|θi〉 = [S,G]|θi〉+ λ−1i G|θi〉, (2.8)

and i = 1, 2, . . . , n. Assuming Λ = diag(λ1, . . . , λn), the equations (2.8) can be written in matrixform as

∂xΘ = ∂xSΘΛ, ∂tΘ = [S,G]Θ +GΘΛ−1.

The Darboux transformation of the generalized coupled dispersionless integrable system in termsof particular matrix solution Θ with the particular eigenvalue matrix Λ is given as

ψ̃ =(λ−1I −ΘΛ−1Θ−1

)ψ, S̃ = S −ΘΛ−1Θ−1, G̃ = G.

In terms of quasideterminants we can write the above expressions as

ψ̃ =

∣∣∣∣∣ Θ ψΘΛ−1 λ−1ψ∣∣∣∣∣ , S̃ = S +

∣∣∣∣ Θ IΘΛ−1 O∣∣∣∣ ,

where O is an n × n null matrix. The result can be generalized to obtain K-fold Darbouxtransformation on matrix solution ψ and can be written in terms of quasideterminant as1 (for

1The quasideterminant for an N ×N matrix over a ring R is defined as

|X|ij =

∣∣∣∣∣ Xij c ijr ji xij∣∣∣∣∣ = xij − r ji (Xij)−1c ij ,

where for 1 ≤ i, j ≤ N , r ji is the row matrix obtained by removing jth entry of X from the ith row. Similarly, cij

is the column matrix containing jth column of X without ith entry. There exist N2 quasideterminants denotedby |X|ij for i, j = 1, . . . , N. For various properties and applications of quasideterminants in the theory of integrablesystems, see e.g. [9, 10, 11, 12, 28].

4 B. Haider and M. Hassan

more details see [16])

ψ[K + 1] = ψ[K]−Θ[K]Λ−1K Θ[K]−1ψ[K] =

∣∣∣∣∣∣∣∣∣∣Θ1 · · · ΘK ψ

Θ1Λ−11 · · · ΘKΛ

−1K λ

−1ψ...

. . ....

...

Θ1Λ−K1 · · · ΘKΛ

−KK λ

−Kψ

∣∣∣∣∣∣∣∣∣∣.

The expression for S[K + 1] is given as

S[K + 1] = S −K∑l=1

Θ[K]Λ−1K Θ[K]−1 = S +

∣∣∣∣∣∣∣∣∣∣∣∣

Θ1 · · · ΘK O...

. . ....

...

Θ1Λ−(K−2)1 · · · ΘKΛ

−(K−2)K O

Θ1Λ−(K−1)1 · · · ΘKΛ

−(K−1)K I

Θ1Λ−K1 · · · ΘKΛ

−KK O

∣∣∣∣∣∣∣∣∣∣∣∣.

The K-fold Darboux transformation on the matrix solution ψ can also be expressed in terms ofHermitian projectors P [K], i.e.

ψ[K + 1] =K∏k=0

(I − µK−k+1 − µ̄K−k+1

λ−1 − µ̄K−k+1P [K − k + 1]

)ψ,

where the Hermitian projection in this case is

P [k] =n∑i=1

|θi[k]〉〈θi[k]|〈θi[k]|θi[k]〉

, k = 1, 2, . . . ,K, (2.9)

with P †[K] = P [K] and P 2[K] = P [K].Now we define the adjoint Darboux transformation. The equation of motion (1.2) and zero

curvature condition (1.5) can also be written as compatibility condition of the following linearsystem (the adjoint Lax pair)

∂xφ = −ξ∂xS†φ, ∂tφ = −[S†, G†

]φ− ξ−1G†φ, (2.10)

which is obtained by taking the formal adjoint of the system (1.4). Note that in equation (2.10)ξ is a real (or complex) parameter and φ is an invertible n×n matrix in the space V† = {φ}. TheDarboux matrix D(ξ) transforms the matrix solution φ in space Ṽ† to a new matrix solution φ̃in Ṽ†, i.e.

D(ξ) : V† → Ṽ†. (2.11)

The one-fold Darboux transformation on the matrix solution φ is defined as

φ̃ ≡ D(ξ)φ = −(ξ−1I − ΩΞΩ−1

)φ,

where Ξ = diag(ξ1, . . . , ξn) is the eigenvalue matrix. The matrix function Ω is an invertiblenon-degenerate n× n matrix and is given by

Ω = (φ(ξ1)|1〉, . . . , φ(ξn)|n〉) = (|ρ1〉, . . . , |ρn〉).

The K-fold Darboux transformation on matrix solutions φ, S† and G† can be expressed as

φ[K + 1] =

∣∣∣∣∣∣∣∣∣∣Ω1 · · · ΩK φ

Ω1Ξ−11 · · · ΩKΞ

−1K ξ

−1φ...

. . ....

...

Ω1Ξ−K1 · · · ΩKΞ

−KK ξ

−Kφ

∣∣∣∣∣∣∣∣∣∣,

Quasi-Grammian Solutions of the Generalized Coupled Dispersionless Integrable System 5

S† [K + 1] = S† +

∣∣∣∣∣∣∣∣∣∣∣∣

Ω1 · · · ΩK O...

. . ....

...

Ω1Ξ−(K−2)1 · · · ΩKΞ

−(K−2)K O

Ω1Ξ−(K−1)1 · · · ΩKΞ

−(K−1)K I

Ω1Ξ−K1 · · · ΩKΞ

−KK O

∣∣∣∣∣∣∣∣∣∣∣∣, G†[K + 1] = G.

In terms of the Hermitian projector we write the above expression as

φ[K + 1] =K∏k=0

(I − νK−k+1 − ν̄K−k+1

ξ−1 − ν̄K−k+1P [K − k + 1]

)φ,

and the Hermitian projector in this case is defined as

P [k] =n∑i=1

|ρi[k]〉〈ρi[k]|〈ρi[k]|ρi[k]〉

, k = 1, 2, . . . ,K. (2.12)

By making use of equations (1.4) and (2.10) for the column solutions |θi〉 and the row so-lutions 〈ρi| of the direct and adjoint Lax pair respectively, it can be easily shown that theexpressions (2.9) and (2.12) are equivalent.

3 Standard binary Darboux transformation

To define the binary transformation we follow the approach of [13, 40, 41, 42, 43, 45] and considera space V̂, which is a copy of the direct space V and the corresponding solutions are ψ̂ ∈ V̂.Since it is a copy of the direct space, therefore the linear system, equation of motion and thezero curvature condition will have the similar form as given for the direct space. The equationof motion (1.2) and zero curvature condition (1.5) can also be written as the compatibilitycondition of the following linear system for the matrix solution ψ̂

∂xψ̂ = Û(x, t, λ)ψ̂, ∂tψ̂ = V̂ (x, t, λ)ψ̂, (3.1)

where

Û(x, t, λ) = λ∂xŜ, V̂ (x, t, λ) =[Ŝ, Ĝ

]+ λ−1Ĝ.

We have taken the specific solutions Θ, Ω for the direct and adjoint spaces V and V† respectively.The corresponding solutions for V̂ are Θ̂ ∈ V̂ and φ̂ ∈ V̂†. Also assuming that i(Θ̂) ∈ Ṽ†, thenfrom equations (2.3) and (2.11), we write the transformation as

D(−1)†(λ) : V† −→ Ṽ†.

Since φ ∈ V†, we have

i(Θ̂) = D(−1)†(λ)φ.

Also from D†(λ)(i(Θ)) = 0, we obtain i(Θ) = Θ(−1)†

and similarly i(Θ̂) = Θ̂(−1)†

. Therefore weget from above equation

Θ̂(−1)†

= D(−1)†(λ)φ, Θ̂ =(D(−1)†(λ)φ

)(−1)†.

By using (2.2) and (2.7) in above equation

Θ̂ =((λ−1I −ΘΛ−1Θ−1

)(−1)†φ)(−1)†

=(λ−1I −ΘΛ−1Θ−1

)φ(−1)

6 B. Haider and M. Hassan

= Θ(λ−1I − Λ−1

)Θ−1φ(−1)† = Θ

(λ−1I − Λ−1

)(φ†Θ

)−1= Θ∆−1, (3.2)

where the potential ∆ is defined as

∆(ψ, φ) =(φ†Θ

)(λ−1I − Λ−1

)−1. (3.3)

Similarly for adjoint space

∧Ω = Ω∆(−1)†,

we obtain

∆(ψ,Ω) = −(λ−1I − Ξ(−1)†

)−1(Ω†ψ

). (3.4)

By writing equations (3.3) and (3.4) in matrix form for the solutions Θ and Ω, we get thefollowing condition on ∆

Ξ(−1)†∆(Θ,Ω)−∆(Θ,Ω)Λ−1 = Ω†Θ, (3.5)

where ∆ is a matrix. An entry ∆ij from equations (3.3), (3.4) and (3.5) is given as

∆(Θ,Ω)ij =(Ω†Θ)ij

ξ̄−1i − λ−1j

. (3.6)

Now we define the Darboux matrix in hat space as

D̂(λ) ≡(λ−1I − Ŝ

)=(λ−1I − Θ̂Ξ(−1)†Θ̂−1

), (3.7)

where

D̂(λ)ψ̂ = ψ̃.

We may summarize the above formulation as

D(λ) : V −→ Ṽ,D̂(λ) : V̂ −→ Ṽ, (3.8)D(ξ) : V† −→ Ṽ†.

The effect of D̂(λ) is such that it leaves the linear system (3.1) invariant, i.e.,

∂x˜̂ψ =

˜̂U(x, t, λ)

˜̂ψ, ∂t

˜̂ψ =

˜̂V (x, t, λ)

˜̂ψ,

where˜̂U and

˜̂V are given as˜̂

U(x, t, λ) = λ∂x˜̂S,

˜̂V (x, t, λ) =

[˜̂S,˜̂G]

+ λ−1˜̂G,

and S̃ and G̃ are the Lie algebra valued transformed matrix fields. By substituting equation (2.1)in equations (2.4), we get

˜̂S = Ŝ − M̂, ˜̂G = Ĝ.

As mentioned earlier equation (2.6) shows that Ĝ is a constant matrix and the matrix M issubjected to satisfy the following equations

∂xM̂M̂ =[∂xŜ, M̂

], ∂tM̂ =

[[Ŝ, Ĝ

], M̂]

+[Ĝ, M̂

]M̂.

Quasi-Grammian Solutions of the Generalized Coupled Dispersionless Integrable System 7

The matrix M can be written in terms of the solutions of the linear system

M̂ = Θ̂Ξ(−1)†Θ̂−1,

and the Darboux transformation on the matrix fields ψ̂ and Ŝ in hat space V̂ is

˜̂ψ =

(λ−1I − Θ̂Ξ(−1)†Θ̂−1

)ψ̂,

˜̂S = Ŝ − Θ̂Ξ(−1)†Θ̂−1.

From equation (3.8) we know that

D̂(λ)ψ̂ = D(λ)ψ,

which implies

ψ̂ = D̂−1(λ)D(λ)ψ. (3.9)

The equation (3.9) relates the two solutions ψ and ψ̂. This transformation is known as thestandard binary Darboux transformation and we write it as B(λ) = D̂−1(λ)D(λ), i.e.

ψ̂ = D̂−1(λ)D(λ)ψ = B(λ)ψ. (3.10)

By substituting (3.7), (2.2) in equation (3.10), we obtain the explicit transformation on ψ as

ψ̂ =(λ−1I − Θ̂Ξ(−1)†Θ̂−1

)−1(λ−1I −ΘΛ−1Θ−1

)ψ

= Θ̂(λ−1I − Ξ(−1)†

)−1Θ̂−1Θ

(λ−1I − Λ−1

)Θ−1ψ. (3.11)

By using (3.2) in equation (3.11), the expression of ψ̂ may be simplified as

ψ̂ = Θ∆(Θ,Ω)−1(λ−1I − Ξ(−1)†

)−1∆(Θ,Ω)Θ−1Θ

(λ−1I − Λ−1

)Θ−1ψ

= Θ∆(Θ,Ω)−1(λ−1I − Ξ(−1)†

)−1∆(Θ,Ω)

(λ−1I − Λ−1

)Θ−1ψ

= Θ∆(Θ,Ω)−1(λ−1I − Ξ(−1)†

)−1(λ−1∆(Θ,Ω)−∆(Θ,Ω)Λ−1

)Θ−1ψ.

By substituting the value of ∆(Θ,Ω)Λ−1 from (3.5), we get

ψ̂ = Θ∆(Θ,Ω)−1(λ−1I − Ξ(−1)†

)−1(λ−1∆(Θ,Ω)− Ξ(−1)†∆(Θ,Ω) + Ω†Θ

)Θ−1ψ

=(I + Θ∆(Θ,Ω)−1

(λ−1I − Ξ(−1)†

)−1Ω†)ψ =

(I −Θ∆(Θ,Ω)−1∆(·,Ω)

)ψ

= ψ −Θ∆(Θ,Ω)−1∆(ψ,Ω), (3.12)

where we have used equation (3.4) in obtaining the last step. Equation (3.12) may be writtenin terms of quasideterminant as

ψ̂ =

∣∣∣∣∣ ∆ (Θ,Ω) ∆ (ψ,Ω)Θ ψ∣∣∣∣∣ . (3.13)

The quasideterminant (3.13) is referred to as quasi-Grammian solution of the system. Theadjoint binary transformation for φ̂ ∈ V̂† is obtained in a simmilar way and gives

φ̂ = φ− Ω∆(Θ,Ω)(−1)†∆†(Θ, φ) =

∣∣∣∣∣ ∆†(Θ,Ω) ∆†(Θ, φ)Ω φ∣∣∣∣∣ .

Again from equation (3.9), we have

Ŝ − Θ̂Ξ(−1)†Θ̂−1 = S −ΘΛ−1Θ−1,

8 B. Haider and M. Hassan

Ŝ = S −ΘΛ−1Θ−1 + Θ̂Ξ(−1)†Θ̂−1 = S −ΘΛ−1Θ−1 + Θ∆(Θ,Ω)−1Ξ(−1)†∆(Θ,Ω)Θ−1.

By using equation (3.5) for Ξ(−1)†∆ (Θ,Ω) in above equation

Ŝ = S −ΘΛ−1Θ−1 + Θ∆(Θ,Ω)−1(∆(Θ,Ω)Λ−1 + Ω†Θ

)Θ−1

= S + Θ∆(Θ,Ω)−1Ω† = S −∣∣∣∣ ∆(Θ,Ω) Ω†Θ O

∣∣∣∣ .For the next iteration of binary Darboux transformation, we take Θ1, Θ2 to be two particularsolutions of the Lax pair (1.4) at Λ = Λ1 and Λ = Λ2 respectively. Similarly Ω1, Ω2 are twoparticular solutions of the Lax pair (2.10) at Ξ = Ξ1 and Ξ = Ξ2. Using the notation ψ[1] = ψ,S[1] = S and ψ[2] = ψ̂, S[2] = Ŝ, we write two-fold binary Darboux transformation on ψ as

ψ[3] = ψ[2]−Θ[2]∆(Θ[2],Ω[2])−1∆(ψ[2],Ω[2]), (3.14)

where Θ[1] = Θ1, Ω[1] = Ω1, Θ[2] = ψ[2]|ψ→Θ2 , Ω[2] = φ[2]|φ→Ω2 . Also note that by using thedefinition of the potential ∆ and equation (3.6), we have

∆(ψ[2], φ[2]) = ∆(ψ1, φ1)−∆(Θ1, φ1)∆(Θ1,Ω1)−1∆(ψ1,Ω1)

=

∣∣∣∣∣ ∆(Θ1,Ω1) ∆(ψ,Ω1)∆(Θ1, φ) ∆(ψ, φ)∣∣∣∣∣ . (3.15)

The equation (3.15) implies that

∆(Θ[2],Ω[2]) = ∆(Θ2,Ω2)−∆(Θ1,Ω2)∆(Θ1,Ω1)−1∆(Θ2,Ω1)

=

∣∣∣∣∣ ∆(Θ1,Ω1) ∆(Θ2,Ω1)∆(Θ1,Ω2) ∆(Θ2,Ω2)∣∣∣∣∣ . (3.16)

By using equations (3.15), (3.16) and the notation defined above in equation (3.14), we get

ψ[3] =

∣∣∣∣∣ ∆(Θ1,Ω1) ∆(ψ,Ω1)Θ1 ψ∣∣∣∣∣−∣∣∣∣∣ ∆(Θ1,Ω1) ∆(Θ2,Ω1)Θ1 Θ2

∣∣∣∣∣×

∣∣∣∣∣ ∆(Θ1,Ω1) ∆(Θ2,Ω1)∆(Θ1,Ω2) ∆(Θ2,Ω2)∣∣∣∣∣−1 ∣∣∣∣∣ ∆(Θ1,Ω1) ∆(ψ,Ω1)∆(Θ1, φ) ∆(ψ,Ω2)

∣∣∣∣∣=

∣∣∣∣∣∣∆(Θ1,Ω1) ∆(Θ2,Ω1) ∆(ψ,Ω1)∆(Θ1,Ω2) ∆(Θ2,Ω2) ∆(ψ,Ω2)

Θ1 Θ2 ψ

∣∣∣∣∣∣ , (3.17)where we have used the noncommutative Jacobi identity2 in obtaining (3.17). The Kth iterationof binary Darboux transformation leads to

ψ[K + 1] = ψ[K]−Θ[K]∆(Θ[K],Ω[K])−1∆(ψ[K],Ω[K])

=

∣∣∣∣∣ ∆(Θ[K],Ω[K]) ∆(ψ[K],Ω[K])Θ[K] ψ[K]∣∣∣∣∣

2For quasideterminants, the noncommutative Jacobi identity is given as∣∣∣∣∣∣E F GH A B

J C D

∣∣∣∣∣∣ =∣∣∣∣ E GJ D

∣∣∣∣− ∣∣∣∣ E FJ C∣∣∣∣ ∣∣∣∣ E FH A

∣∣∣∣−1 ∣∣∣∣ E GH B∣∣∣∣ .

For the definition and more properties of quasideterminants see e.g. [9, 10, 11, 12, 28].

Quasi-Grammian Solutions of the Generalized Coupled Dispersionless Integrable System 9

=

∣∣∣∣∣∣∣∣∣∆(Θ1,Ω1) · · · ∆(ΘK ,Ω1) ∆(ψ,Ω1)

... · · ·...

...∆(Θ1,ΩK) · · · ∆(ΘK ,ΩK) ∆(ψ,ΩK)

Θ1 · · · ΘK ψ

∣∣∣∣∣∣∣∣∣ . (3.18)Above result can be proved by induction by using the properties of quasideterminants. Similarlythe Kth iteration of adjoint binary Darboux transformation gives

φ[K + 1] = φ[K]− Ω[K]∆(Θ[K],Ω[K])(−1)†∆(Θ[K], φ[K])†

=

∣∣∣∣∣ ∆(Θ[K],Ω[K])† ∆(Θ[K], φ[K])†Ω[K] φ[K]∣∣∣∣∣

=

∣∣∣∣∣∣∣∣∣∣∣

∆(Θ1,Ω1)† ∆(Θ2,Ω1)

† · · · ∆(ΘK ,Ω1)† ∆(Θ1, φ)†∆(Θ1,Ω2)

† ∆(Θ2,Ω2)† · · · ∆(ΘK ,Ω2)† ∆(Θ2, φ)†

...... · · ·

......

∆(Θ1,ΩK)† ∆(Θ2,ΩK)

† · · · ∆(ΘK ,ΩK)† ∆(ΘK , φ)†

Ω1 Ω2 · · · ΩK φ

∣∣∣∣∣∣∣∣∣∣∣.

The multisoliton S[K+1] can be obtained by putting λ = 0 in the expression for ψ[K+1] (3.18)and using G = ψ|λ=0, which on silmplification gives

S[K + 1] = S −

∣∣∣∣∣∣∣∣∣∆(Θ1,Ω1) · · · ∆(ΘK ,Ω1) Ω†1

... · · ·...

...

∆(Θ1,ΩK) · · · ∆(ΘK ,ΩK) Ω†KΘ1 · · · ΘK I

∣∣∣∣∣∣∣∣∣ .Similar expression can be obtained for the Kth iteration of S†.

Therefore by using the standard binary Darboux transformation we have obtained the gram-mian type solutions for the linear system and the potential is also expressed in terms of quaside-terminants. That is by constructing binary Darboux transformation in terms of spectral para-meter we can get the expression of the matrix solution of the linear system in terms of grammiantype quasideterminants which is different in representation from the solutions obtained by ele-mentary Darboux transformation. In addition to the solutions of the linear system we are alsoable to obtain explicit quasideterminant expression of the potential ∆ in terms of the particularsolutions of the linear system. It is important to note that the spectral parameter remains un-changed in binary Darboux transformation. We consider the eigenfunctions (solutions of directLax pair) and adjoint eigenfunctions (solutions of adjoint pair). The bilinear potential ∆ isrelated to each pair of (direct and adjoint) solutions. Since we know that the solutions of thelinear system can be column vectors or they can be combined to give solution in matrix form.As in the present case when the solutions are in matrix form the potential ∆ is also a matrix.It has been shown earlier that matrix solutions can be reduced to vector solutions [15]. In sucha case when solutions are vectors the potential ∆ becomes scalar and by replacing spectral para-meter with derivative we can consider potential to be a contour integration of the correspondingexpressions in x − t plane. In the next section we will see what happens when we apply ourmethod to a specific case of SU(2) system.

4 Explicit solutions for the SU(2) system

In this section we consider the generalized coupled dispersionless integrable system based on theLie group SU(2) and calculate the soliton solutions by using binary Darboux transformation.

10 B. Haider and M. Hassan

For the Lie group SU(2) the matrix fields S and G are valued in the Lie algebra su(2) and wehave

S† = −S, G† = −G, (4.1)TrS = 0, TrG = 0. (4.2)

Following the same steps for the direct Lax pair as obtained in [16]. We define a vector φ =(φ1, φ2, φ3) in such a way that the matrix field S is given by

S = i

(φ3 φ1 − iφ2

φ1 + iφ2 −φ3

). (4.3)

Equation (4.3) satisfies the conditions (4.1) and (4.2). The matrices U and V are then given as

U = iλ

(∂xφ3 ∂xφ1 − i∂xφ2

∂xφ1 + i∂xφ2 −∂xφ3

), V =

(0 φ1 − iφ2

−φ1 − iφ2 0

)− i

2λ

(1 00 −1

).

By writing φ1 = r, φ2 = 0 and φ3 = q, we get the coupled dispersionless integrable system asgiven in [24]

∂x∂tq + 2∂xrr = 0, ∂x∂tr − 2∂xqr = 0.

and the matrix S from equation (4.3) is given as

S = i

(q rr −q

). (4.4)

To obtain the expression for the Darboux matrix we take

Λ =

(λ1 00 −λ1

), Θ =

(α ββ −α

). (4.5)

By using equations (4.5) and (2.7) in equation (2.2), we get

D(λ) =

(λ−1 − λ−11 cosω −λ

−11 sinω

−λ−11 sinω λ−1 − λ−11 cosω

),

where we have assumed that tan ω2 =βα . We now consider the seed solution as follows

ψ =

(eiλx−

i2λt 0

0 e−iλx+i2λt

). (4.6)

By using the above equation (4.6) we can write the particular matrix solution Θ of the directLax pair (1.4) as

Θ =(ψ(λ1)|1〉 ψ(λ2)|2〉

)=

(eiλ1x− i2λ1 t e

iλ2x− i2λ2 t

e−iλ1x+ i2λ1 t −e−iλ2x+

i2λ2

t

). (4.7)

By substituting λ2 = −λ1 we get from above equation (4.7)

Θ =

(eiλ1x− i2λ1 t e

−iλ1x+ i2λ1 t

e−iλ1x+ i2λ1 t −eiλ1−

i2λ1

t

). (4.8)

Quasi-Grammian Solutions of the Generalized Coupled Dispersionless Integrable System 11

Taking l = 2iλ1x− iλ1 t and using the definition (2.7) we get from (4.8)

M =λ−11

2 cosh l

(2 sinh l 2

2 −2 sinh l

)= λ−11

(tanh l sech lsech l − tanh l

). (4.9)

From equation (2.5) we have

∂xS[1] = ∂xS − ∂xM.

On comparison with equation (4.4) and using (4.9) we obtain

∂xq[1] = ∂xq + i∂xM11 = 1 + iλ−11 ∂x tanh l = 1− 2 sech

2 l

= 1− 2 sech2(

2iλ1x−i

λ1t

), (4.10)

where we have used ∂xq = 1. Similarly we have for r = 0

∂xr[1] = ∂xr + i∂xM12,

which gives

r[1] = iM12 = iλ−11 sech

(2iλ1x−

i

λ1t

). (4.11)

It is easy to show from equations (4.10) and (4.11) that in the asymptotic limit ∂xq[1]→ 1 andr[1]→ 0. Now by making use of above calculation we can write the iterated solution ψ[1] as

ψ[1] = D(λ)ψ =

((λ−1 − λ−11 tanh l

)el2 −λ−11 sech le

−l2

−λ−11 sech el2

(λ−1 + λ−11 tanh l

)e−l2

).

Repeating the calculations as we did for direct pair, we get

Ω =

(ep2 e−

p2

e−p2 −e

p2

), (4.12)

where p = 2iξ1x − iξ1 t. To obtain the expression for Ŝ, we start with the definition (3.6) of∆(Θ,Ω), ξ̄ = −ξ and by using (4.8), (4.12) obtain for the present case

∆(Θ,Ω) =

− 2 cosh l̂ξ−11 + λ

−11

− 2 sinh p̂ξ−11 − λ

−11

− 2 sinh p̂ξ−11 − λ

−11

2 cosh l̂

ξ−11 + λ−11

,where

l̂(x+, x−) = i(ξ1 + λ1)x−i

2

(1

ξ1+

1

λ1

)t, p̂(x+, x−) = i(ξ1 − λ1)x−

i

2

(1

ξ1− 1λ1

)t.

Now we consider

M̂ = Θ∆(Θ,Ω)−1Ω† =

(M̂11 M̂12M̂21 M̂22

)

12 B. Haider and M. Hassan

=4

K

cosh l̂ sinh l̂

ξ−11 + λ−11

+cosh p̂ sinh p̂

ξ−11 − λ−11

cosh l̂ cosh p̂

ξ−11 + λ−11

− sinh p̂ sinh l̂ξ−11 − λ

−11

cosh l̂ cosh p̂

ξ−11 + λ−11

− sinh p̂ sinh l̂ξ−11 − λ

−11

−cosh l̂ sinh l̂ξ−11 + λ

−11

− cosh p̂ sinh p̂ξ−11 − λ

−11

, (4.13)where

K = det ∆(Θ,Ω) =−4 cosh2 l̂(ξ−11 + λ

−11

)2 − 4 sinh2 p̂(ξ−11 − λ

−11

)2 , Ŝ = S + Θ∆(Θ,Ω)−1Ω†,(iq[1] ir[1]ir[1] −iq[1]

)=

(iq + M̂11 ir + M̂12ir + M̂21 −iq + M̂22

)(4.14)

From equation (4.14) and (4.13) by using ∂xq = 1 and r = 0 we get

∂xq[1] = 1 +8λξ

K

[sinh p sinh l +

2

K

{sinh 2l̂

ξ−11 + λ−11

− sinh 2p̂ξ−11 − λ

−11

}], (4.15)

r[1] = −i 4K

{cosh l̂ cosh p̂

ξ−11 + λ−11

− sinh p̂ sinh l̂ξ−11 − λ

−11

}. (4.16)

In the asymptotic limit for t → ±∞, we have l̂ → ±∞ and the equations (4.15) and (4.16)become

liml̂→±∞

∂xq[1] = 1, liml̂→±∞

r[1] = 0. (4.17)

We see that in the asymptotic limit, we get much simpler expressions. Note that the expressionis similar to the one we obtain from elementary Darboux transformation. Now we consider thespecial case when ξ = λ which gives p̂ = 0. The solutions (4.15) and (4.16) become

∂xq[1] = 1 + 2 sech2 l̂, (4.18)

r[1] = iλ−1 sech l̂. (4.19)

On comparison of equations (4.10) and (4.11) with equations (4.15) and (4.16) we see that theoriginal solutions obtained by the standard binary Darboux transformation are different fromthose of elementary Darbouix transformation and contain the contribution from both the directand adjoint system. If we take ξ = λ the solutions from both the techniques become equal asshown by equations (4.18) and (4.19). Therefore the advantage of using standard binary Darbouxtransformation is that we can obtain the solutions in the form of direct and adjoint spaceparameters and then without using elementary Darboux transformation we can obtain solutionsjust by equating parameters as shown above where we have obtained equations (4.18) and (4.19)(which have same form as solutions obtained from elementary Darboux transformation) fromequations (4.15) and (4.16) (which give solutions by standard binary Darboux transformation).

It is simple to show that the solutions (4.18) and (4.19) have the same behaviour in asymptoticlimit and satisfy (4.17). Plots of solutions (4.18) and (4.19) for λ = i are shown in Figs. 1 and 2.Now we show the relationship between our solution of the system (1.3) and the solution φ ofthe sine-Gordon equation. The sine-Gordon equation is given as

∂x∂tφ = 2 sinφ,

and is related to our system by the following equations

∂xq = cosφ, r = ±1

2∂tφ. (4.20)

Quasi-Grammian Solutions of the Generalized Coupled Dispersionless Integrable System 13

Figure 1. Plot of solution (4.18) representing

one soliton solution ∂xq[1].

Figure 2. Plot of solution (4.19) representing

one soliton solution r[1].

To obtain the expression for φ[1], we use equation (4.20) in equation (4.19) which gives

±12∂tφ[1] = iλ

−1 sech l̂,

φ[1] = ±2iλ−1∫

sech

(2iλx− i

λt

)dt = ± 2

λ2tan−1

(exp

(2iλx− i

λt

)). (4.21)

The equation (4.21) is the one-kink solution to the sine-Gordon equation [33].

5 Conclusions

In this paper, we have composed the elementary Darboux transformations of the generalizedcoupled dispersionless system and obtained the standard binary Darboux transformation ofthe model. By iterating the standard binary Darboux transformation we have generated themultisolitons of the model. We have also obtained the quasideterminant expression for thepotential ∆. We have also considered the case of coupled dispersionless integrable system basedon the Lie group SU (2), and have obtained explicit expressions of Grammian solutions of thesystem. There are various directions in which the the integrability properties of the generalizeddispersionless integrable system can be studied. One such study is to investigate the r-matrixstructure and the existence of infinitely many conservation laws of the system. We shall returnthese and related investigations in a separate work.

Acknowledgements

BH would like to thank Department of Physics, University of the Punjab, Lahore, Pakistan forproviding the research facilities.

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1 Introduction2 Darboux transformation on the direct and adjoint Lax pairs3 Standard binary Darboux transformation4 Explicit solutions for the SU(2) system5 ConclusionsReferences