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Lie Algebraic Structures and Integrability of Long-Short Wave Equation in (2+1)Dimensions

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2004 Commun. Theor. Phys. 42 821

(http://iopscience.iop.org/0253-6102/42/6/821)

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Commun. Theor. Phys. (Beijing, China) 42 (2004) pp. 821–823c© International Academic Publishers Vol. 42, No. 6, December 15, 2004

Lie Algebraic Structures and Integrability of Long-Short Wave Equation in (2+1)Dimensions∗

ZHAO Xue-Qing and LU Jing-FaCollege of Physics, Nankai University, Tianjin 300071, China

(Received March 2, 2004)Abstract The hidden symmetry and integrability of the long-short wave equation in (2+1) dimensions are consideredusing the prolongation approach. The internal algebraic structures and their linear spectra are derived in detail whichshow that the equation is integrable.

PACS numbers: 02.09Key words: prolongation approach, hidden symmetry, integrability

1 IntroductionDue to the importance of equations describing the in-

teractions between the long and short waves in fluid me-chanics, there have been many studies on these equationsin one or two spatial dimensions.[1−5] N. Yajima[1] andY.C. Ma[2] showed that the nonlinear evolution equationsin one spatial dimension are exactly solvable by the inversescattering method and they gave an N -soliton solution de-caying at infinity. Recently, M. Oikawa et al.[5] derived thefollowing long-short wave resonance equation (LSWE) fora two-layer fluid model in (2 + 1) dimensions,

i(St + Sy)− Sxx + LS = 0 ,

Lt = (2SS∗)x; Lt = ∂L/∂t ,

Sxx = ∂2S/∂x2, . . . , (1)

where L and S denote the long interfacial waves and theshort surface waves packets, respectively. They obtainedtwo types (bright and dark) of soliton solutions using Hi-rota’s bilinear equation method. Later on, D.W.C. Lai etal.[6] have shown that equation (1) possesses “position”and “dromion” solutions. So it would seem that equa-tion (1) belongs to an integrable class. In order to justifythis conjecture, we discuss the internal algebraic struc-tures and the integrability of Eq. (1) using a differentialgeometrical method in the spirit of Wahlquist–Estabrook,namely the prolongation approach.[7] The use of prolon-gation approach on evolution equations in two spatial di-mensions was first performed by H. Morris while analyz-ing the inverse scattering problem of KPD equation andthe generalized AKNS equations.[8] We will introduce anenlarged prolongation 2-form to discuss the internal al-gebraic structure of Eq. (1) with the aim to analyze itsintegrability.[9] The linear spectra of Eq. (1) is obtainednaturally using our approach.

2 Modified Equations in (1+1) Dimensionsand Its prolongation StructuresTo generalize the prolongation approach for an evolu-

tion equation in one spatial dimension to a general system

of equations in two spatial dimensions, we first discuss amodified system of equations associated with Eq. (1) andstudy the prolongation structure of a (1 + 1) dimensionalsystem related to these modified equations. First of all,we separate the real and imaginary parts of the functionS in Eq. (1),

S = f + ig , (2)

where f and g are both real-valued functions. Substitut-ing Eq. (2) into Eq. (1), we obtain the following modifiedequations

Lt = 2ffx + 2ggx ,

ft = −fy + gxx − Lg ,

gt = −gy − fxx + Lf . (3)

The corresponding equations in (1 + 1) dimensions willthen be obtained from Eq. (3) by setting ∂y = 0. Thisleads to

Lt = 2ffx + 2ggx ,

ft = gxx − Lg ,

gt = −fxx + Lf . (4)

Related to these equations, the following 2-forms, i.e.

α1 = df ∧ dt− πdx ∧ dt ,

α2 = dg ∧ dt− ηdx ∧ dt ,

α3 = dL ∧ dx + (2fπ + 2gη)dx ∧ dt ,

α4 = df ∧ dx + dη ∧ dt− Lgdx ∧ dt ,

α5 = dg ∧ dx− dπ ∧ dt + Lf dx ∧ dt , (5)

are written in such a way that they are the Pfaff form ofEq. (4). They will become null on the solution manifoldof Eq. (4) and therefore generate a closed ideal.

In general, we may introduce an enlarged prolongation1-form to describe the linear prolongation structure of any

∗The project supported in part by National Natural Science Foundation of China and the Natural Science Foundation of Henan Province

under Grant No. 20011400015

822 ZHAO Xue-Qing and LU Jing-Fa Vol. 42

evolution equation in (1 + 1) dimensions. The enlarged 1-form can be written as

Ωβ =M∑

α=1

(F βα dx + Gβ

αdt)ξα + dξβ , β = 1, 2, . . . ,M , (6)

dΩβ =N∑

i=1

fβiαi +M∑

γ=1

ηβγ ∧ Ωγ , i = 1, 2, . . . , N , (7)

where F and G are some m×m matrix functionals of f , g,L, π (π = fx), and η (η = gx). The integrability conditionis defined by

dΩ ∈ I(Ω, α) , (8)

where I(Ω, α) is an closed ideal generated by the set ofαi and Ωβ. Taking the exterior derivatives of Eq. (6)and substituting Eq. (5) into the results, we obtain a set ofpartial differential equations for F and G on the solutionmanifold,

∂F

∂π= 0,

∂F

∂η= 0,

∂G

∂L= 0 ,

∂G

∂η=

∂F

∂f,

∂G

∂π= −∂F

∂g,

[F,G] + Lg∂F

∂f− Lf

∂F

∂g− 2(fπ + gη)

∂F

∂L

+ π∂G

∂f+ η

∂G

∂g= 0 . (9)

Solving these equations we find

F = X0 + X1f + X2g + X3L ,

G = Y1f + Y2g + Y3f2 + Y4g

2 −X2π + X1η , (10)

where Xi (i = 0, 1, 2, 3) and YJ (j = 1, 2, 3, 4) are someoperators satisfying the following commutation relations,

[X0, Y1] = 0, [X2, Y1] = −[X1, Y2] ,

[X2, X1] = 2X3 − 2Y4 ,

[X0, Y2] = 0, [X1, Y3] = 0, [X3, Y1] = X2 ,

[X1, Y4] = 0, [X0, Y3] = −[X1, Y1], [X3, Y2] = −X1 ,

[X3, Y3] = 0, [X0, Y4] = −[X2, Y2] ,

[X1, X2] = 2Y3 − 2X3 ,

[X2, Y3] = 0, [X0, X2] = Y1, [X3, Y4] = 0 ,

[X2, Y4] = 0, [X0, X1] = −Y2 ,

[X3, X2] = 0, [X3, X1] = 0 . (11)

These relations are henceforth referred to as prolongationstructures. Obviously, they form an incomplete algebrabecause [X0, X3] and [Yi, Yj ] are not yet given. However,we can show that this incomplete algebra can be linearlyembedded into SU(3) Lie algebra as

X0 = T4 + iT5, X1 = −iλ(T1 − iT2) +12λ

(T6 − iT7) ,

X2 = −λ(T1 − iT2) +i

2λ(T6 − iT7), X3 = T4 − iT5 ,

Y1 =i

2λ(T1 + iT2) + λ(T6 + iT7) ,

Y2 = − 12λ

(T1 + iT2)− iλ(T6 + iT7) ,

Y3 = Y4 =12(T4 − iT5) , (12)

where λ is an arbitrary constant and Tm (m = 1, 2, . . . , 8)are generators of SU(3) Lie algebra satisfying

[Tn, Tm] = iεnmkTk . (13)

Therefore, using the standard vector representation of Tm,we can obtain the matrix form of F and G as

F =

0 0 1−iλ(f − ig) 0 0

L12λ

(f + ig) 0

,

G =

0i

2λ(f + ig) 0

λ(π − iη) 0 λ(f − ig)12(f2 + g2) − i

2λ(π + iη) 0

. (14)

These matrices will be used to describe the internal alge-braic structures of the (2+1)-dimensional equation (3).

3 Lie Algebra Structures of (2+1)-Dimen-sional LSWEIn this section, we will study the prolongation struc-

ture for the (2+1)-dimensional system of evolution equa-tions (3). To begin with, we first describe the genericmethod which will be used to construct the prolongationstructure of a system of evolution equations in two spatialdimensions. We need to define the 2-forms

Ωβ = Ωβ ∧ dt +M∑

γ=1

Hβγ ξγ dx ∧ dy

+ (Aβγ dx + Bβ

γ dy) ∧ dξγ , (15)

dΩβ =N∑

i=1

fβiαi +M∑

γ=1

ηβγ ∧ Ωγ , (16)

where A and B are some m×m constant matrices satis-fying the conditions

[A,B] = 0 , (17)

[G, A] + [B,F ] = 0 , (18)

and take a set of N 3-forms αi (i = 1, 2, . . . , N) in theforms

αl = αl ∧ dy, l = 1, 2, . . . , k , (19)

αj = αj ∧ dy + βj , j = k + 1, . . . , N . (20)

Taking the exterior derivatives of Eq. (15) and comparingwith Eq. (16), we get

H = GA− FB , (21)

No. 6 Lie Algebraic Structures and Integrability of Long-Short Wave Equation in (2+1) Dimensions 823

and the following relation which will be used to determinethe parameter βj

N∑j=k+1

fβj βj = [(dGA− dFB)ξ]β ∧ dx ∧ dy . (22)

To use the above general scheme to the (2+1)-dimensional equations (3), we write down their 3-formsαi as

α1 = α1 ∧ dy, α2 = α2 ∧ dy ,

α3 = α3 ∧ dy ,

α4 = α4 ∧ dy − df ∧ dx ∧ dt ,

α5 = α5 ∧ dy + dg ∧ dx ∧ dt . (23)

Comparing the relations (23) to (20), we have

β4 = −df ∧ dx ∧ dt ,

β5 = −dg ∧ dx ∧ dt . (24)

We can also obtain a linear system of inverse scatteringproblem for such an equation on the solution manifold ofEqs. (15) and (16), which turns out to be of the form

ξix = −Fξi −Aξi

y, ξit = −Gξi −Bξi

y , (25)

where the matrices F and G are given by Eq. (14) andξi = (ξ1, ξ2, ξ3). The new constant matrices A and B areyet to be determined.

Substituting Eq. (25) into Eq. (6) and taking accountof Eq. (16), we obtain

f14 = f15 = 0, f25 = −λξ1 ,

f24 = if25, f34 = ξ2/2λ, f35 = if34 . (26)

We assume that the matrices A and B take the follow-ing form

A =

a11 a12 a13

a21 a22 a23

a13 a23 a33

,

B =

b11 b12 b13

b21 b22 b23

b31 b32 b33

. (27)

Then equations (22), (24), and (26) imply that only

A =

0 0 00 0 0i 0 0

, B =

0 0 00 1 00 0 0

(28)

satisfy the relations (17) and (18). According to the ex-pressions (28) and (14), we may rewrite Eq. (25) as

ξ1x = −ξ3, ξ2

x = iλ(f − ig)ξ1,

ξ3x = −Lξ1 − 1

2λ(f + ig)ξ2 − iξ1

y ,

ξ1t = − i

2λ(f + ig)ξ2 ,

ξ2t = −λ(π − iη)ξ1 − λ(f − ig)ξ3 − ξ2

y ,

ξ3t = −1

2(f2 + g2)ξ1 +

i2λ

(π + iη)ξ2 , (29)

which are the linear spectra of the long-short wave equa-tion (1) in (2+1) dimensions.

4 ConclusionsIn this paper, we exhibited that the prolongation ap-

proach extended to two spatial dimensions can be usedto study the internal algebraic structure and the integra-bility of (2+1)-dimensional evolution equations. The pro-longation structure of LSWE in (2+1) dimensions can berealized by SU(3) Lie algebra. We also showed that sucha finite-dimensional algebra gives a linear spectra of anintegrable system. So the prolongation method providesa powerful tool for considering the internal structures ofa dynamic equation with regards to its integrability. Inview of this, it is valuable to develop the relevant theoryfurther, which will be useful for a deeper understandingof the integrability of higher dimensional equations.

References

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[5] M. Oikawa, M. Okamura, and M. Funakoshi, J. Phys. Soc.Jpn. 58 (1989) 4416.

[6] Derek W.C. Lai and K.W. Chow, J. Phys. Soc. Jpn. 68(1999) 1847.

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