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Singularity structure analysis and the complete integrability of the higher order nonlinear Schrödinger-Maxwell-Bloch equations
Chaos, Solitms
& Fractals Vol. 7, No. 3. pp. 377-.3X2. 1996 Copyright 0 1996 Elsevier Science Ltd Prmted in Great Britain. All rights reserved 096&-077Y/Y6 $15.00 + 0.00
0960-0779(95)00069-O
Singularity Structure Analysis and the Complete Integrability of the Higher Order Nonlinear Schriidinger-Maxwell-Bloch Equations* K. PORSEZIAN and K. NAKKEERAN Department of Physics, Anna University, Madras-600025, India (Accepted 4 July 1995)
propose the coupled system of the generalized nonlinear Schrodinger equation and the Maxwell-Bloch equations, which finds application in nonlinear fiber optics. Using the Painleve singularity structure analysis, the condition for the complete integrability is obtained. The Lax pair and other properties of the equations are discussed. With suitable parametric conditions, other interesting soliton models are generated.
Abstract-We
1. INTRODUCTION
Recently there has been considerable interest in the singularity structure analysis of the several physically interesting coupled nonlinear partial differential equations such as, the coupled nonlinear Schrodinger (NLS) equation [ 11, coupled higher order NLS equation [2], coupled Klein-Gordon equation [3] and so on. Nowadays the above analysis can also be used to investigate the coupled integrability properties of given partial differential equations like, the Lax pair, Backlund transformations, bilinear form, etc. Hence, the singularity structure analysis is considered to be one of the most powerful and systematic method in the field of nonlinear science. As large number of articles [4-61 are devoted to this analysis, we do not give a detailed introduction about the method. In this paper, we mainly consider one of the very important problems in the area of nonlinear fiber optics, namely the coupled higher order NLS-Maxwell-Bloch (HNLS-MB) equations in the form,
(1)
where E is the slowly varying envelope of electric field, the subscripts z and t denote the spatial and temporal partial derivatives, p and q are given by v1v2* and /v1I2 - Iv212 respectively. Here y1 and V~are the wave functions of the two energy levels of the resonant atoms. The bracketed term (. . .) means, the averaging over the entire frequency range, co (P(Z,
t; 0))
=
I --a
p(z,
cc
h(w)do= 1, I --a where h(o) is the uncertainty in the energy level. *Communicated by Professor Wadati. 371
W44
da
(2)
378
K. PORSEZIAN and K. NAKKEERAN
Equation (1) when E = 0 reduces to the NLS-MB equations [7],
E, = ~cI&~ + 1~12~1+ 24(p), pt = Ziwp + Eq, 7, = -(Ep*
(3)
+ Ep*),
when N = y = 0 and @= 1, (1) reduces to the derivative NLS-MB equations of the form, E, = ial[kE,
+ (E12E] - E@IE/~E, + 2a2(p),
pI = 2iwp + Ey, qr = -(Ep*
(4)
+ Ep*).
With CY= 0 and /I = y = 1, (1) results in mixed derivative NLS-MB equations which have the form, E, = icul[$,t + lE12E] - E[IEI~E, + E(lE12),1 + 2a~(p), pt = 2iwp + Ey, qt = -(Ep*
(5)
+ Ep”).
For @= 60~~and y = 0, coupled Hirota equation and Maxwell-Bloch (H-MB) can be derived from HNLS-MB equations. H-MB equations take the form, EZ = ial[iE,, pr = 2iop
+ jE12E] - E[E,, + 6jE(‘E,]
equations
+ 2cu,(p),
+ Er],
q, = -(Ep”
(6)
+ Ep*)
Thus it is understood that (1) is in general not integrable but contains several interesting families of equations, having very important applications in the study of the wave propagation in the erbium doped nonlinear fiber media. The physical application of the above papers will be published elsewhere. Except (3), the integrability aspects of the equations (1, 4-6), are still an open question. In order to investigate the integrability aspects of equation (1) we carry out a singularity structure analysis in Section 2, and show that (1) is integrable only for the condition 3a = b = 2y. After analyzing the Painleve property, the explicit Lax pair is also constructed. 2. PAINLEVk
ANALYSIS
To apply Painleve analysis, first one has to remove the averaging or integral function in (1). This can be solved by considering the function h(w) in (1) as a Dirac-delta function at the resonant frequency. Under this assumption the averaging function over the entire frequency range can be eliminated from the variable p. Further, for convenience, independent variables z and t are interchanged. With all these modifications, (1) takes the form, E, = ial[~E,, pz = 2iup rjz = -(Ep*
+ IEj2E] + Ev,
- &[cYYE~~~ + ,GIE12E, + yE(IEi2),]
+ ~CCY,P, (7)
+ Ep”).
The nature of the dependent variables E and p are complex, where as q is a real varip=c, p*=d and q=e able. To apply Painleve analysis, we express E = a, E*=b,
Integrability of the higher order nonlinear Schrbdinger-Maxwell-Bloch
equations
379
(* represents a complex conjugate). With the new dependent variables a, . . ., e, the set of (7) becomes a, = i&2,,
+ a2b) - E[auzzZ+ (/3 + y)aba, + yu2b,] + ~CQC,
b, = -k&b,,
+ ab2) - E[ab,,, + (p + y)abb, + yb’u,]
+ 2azd,
cz = 2ioc + 2ue, d, = -2iwd eZ = -(ad
(8)
+ 2be, + bc).
The generalized Laurant series expansions of a, . . . , e are, U =
W’CUj(ZY
t)q’(Z,
t)l
t)q’(Z>
‘)>
‘)V’(zy
‘)2
t)@(Z,
f).
j=O
@iCj(Z>
C =
(9)
j=O
d = q’Cdj(Z, j=O
e=
qqCej(Z, j=O
With uo, . . ., e. # 0; where 1, m, II, Y and q are negative integers; Uj, . . . , ej are the set of expansion coefficients which are analytic in the neighbourhood of the non-characteristic singular manifold QI(Z, t) = z + v(t) = 0. Looking at the leading order, we substitute u=uOcp’, . . ., e=eoqq in equation (8) and upon balancing terms we obtain the following results, I=
m = -1,
aobo = -1,
n=r=q
(10)
boco = aado.
Substituting the full expansion of the Laurent series and keeping the leading order terms alone, we obtain the following equation, A Bb; -2eo 0 do
Bu:, A 0 -2eo aa
0 (i -f n> 0 bo
0 0 (j +” 4 a0
0 0 -2uo -2bo (j + 4
= 0
(11)
where A = a(j - l)(j
- 2)(j - 3) + (j - l)(a + /3)uobo - (0 + 3cu)uobo,
B = (j - 1) - (p + y). Substituting (10) and solving the determinant the resonance values are found to be, j = -l,O,
3,4, -n, f
112 (174
380
K. PORSEZIAN and K. NAKKEERAN
From careful analysis, we find that (11) admits a sufficient number of positive resonances only when y1= -2 and for the condition 3a = p = 2y which emphasizes that (8) is non-integrable for other values of a, /3 and y. Substituting the values of n, N, /I?and y, the resonance values are found to be, j = -1, 0, 0, 2, 2, 3, 4, 4, 4.
(13) As usual, the resonance value at j = -1 corresponds to the arbitrariness of the singularity manifold 91and j = 0, 0 implies that a0 or b. and co or do are arbitrary which is evident from (10). Similarly from the coefficients of (qm3, qm3, CJ-‘, cp-*, qe2) we obtain,
(14) i EbO(P Ebo(B 2eo 40++ 37) Y)
Eao(p eao(8 2e0 co 0++ 3y) r>
bo 01
a0 01
2bo 2u, -10 i
Solving (14), the values of aI, . . ., el are found to be, al = 0, bl = 0, cl = 2iaoeo,
(15)
d, = -2iboeo, el = 0. Similarly we also obtain the matrix for a2, . . ., e2 as, ’ -2Ey
db:, 2eo 0
&/la: -2&Y 0 2eo
0 0 0 0
0 0 0 0
0 0 2a. 2bo
co
bo
QO
0
, do
a2 b2 c2 4 e2
F -2icr - 2a,e, 2idl - 2b,e, -aId, - b,c,
,
(16)
where D = -u~v,~~- ia,(2aoboal + aibl) - E/h.oa,b, - Ey(aoalb, + a:bo) - ~cY~c~~, F = -boqt + ial(2aobob, + bf,a,) - &fib,a,b, - &y(boalbl + b:ao) - 2a2do. Equation (16) reveals that there are two arbitrarinesses, corresponding to the resonances j = 2, 2. Also f rom the remaining powers of Q?,we find that (8) admits a sufficient number of arbitrary functions only for ~CV= p = 2y. Hence we conclude that the equations in (6) are expected to be integrable only for the above choice of parameters. So, the integrable version of (1) takes the form, E, = ial[$E,
+ /Ej*E] - &[Ettt + 3jEIZE, + $E(IEj*),] + 2a2(p),
pt = 2iop + Eq, qt = -(Ep*
(17)
+ Ep”).
Hence (17) admits the Painleve property and is expected to be integrable only for the above parametric choice. Then to establish the integrability properties, we construct the linear eigenvalue problem in the following form,
Integrability
of the higher order nonlinear
SchrGdinger-Maxwell-Bloch
equations
381
(18)
where
-i- &[(EE; - EXE*) + +]
+ 2icu,
0 -p"e" i,A- io 1
-r i A - -iBiw i pe 1,A- io 1
p*e” i I A- io.1 0
Here 0 = l/6&(2 - t/l&), x = z, - t/12~ and ?, (= A1+ i&) is the spectral parameter, so that the consistency condition Ur, - VI, + [ Ur, VI] = 0 leads to HNLS-MB equations. The construction of the soliton solution is in progress. 3. PAINLEVI?
ANALYSIS
OF THE HIROTA-MB
EQUATIONS
As we have discussed in the introduction, when /II = 6a and y = 0, (1) reduces to Hirota-MB (H-MB) equations. Substituting /3 = 6cx, y = 0 and n = -2 in (12), we get the resonances for the H-MB equations as j = -1, 0, 0, 1, 2, 3, 4, 4, 5. (19) Equation (6) also admits the sufficient number of arbitrary functions. The 2 X 2 linear eigenvalue problem of (6) is Yv, = U,Y, Yv, = V,Y, where
Y = (Y,
Y,)T
cw
K. PORSEZIAN and K. NAKKEERAN
382
Here G
= -4iE/i3
H = 4id’E
-
2ia,h2 + 2id\E)*
+ 2a,AE + 2idE,
+ ia,jE/* + E(EE; - E,E*), - 2&jE12E + ialE,
- EE,.
The H-MB equations are obtained from the consistency condition U,, - V,, + [V,, V,] = 0. Using Blcklund transformations [S], the single soliton solution [9] of the H-MB equation is obtained as E(e,t) = 2k2sech(p)exp(io
- if&),
(21)
where p and o are functions of z, t and soliton velocity parameters given by,
cT(z, t) = 2A,t + 4a,(A: - A:) -
m 2c~~(A~- w)h(o)dw I --cc A; + (A1 - o$
+ 8~Iz~(/tf
-
32;)
1
2 + do),
(p”’ and o(O)are independent of both z and t). And 6r is a real constant. 4. CONCLUSION
Thus, in this paper we have investigated the singularity structure analysis of the HNLS-MB equations which describes the wave propagation in erbium doped nonlinear fiber media. From the detailed analysis, we find the system admits Painleve property only for a particular condition between the parameters involved in the equations. We also discussed the other integrable systems involved in our proposed model. The Lax pair is explicitly constructed. Acknowledgements-K. Porsezian wishes to thank the Council of Scientific and Industrial Research and the Department of Science and Technology, Government of India, for their financial assistance through projects. The author K. Nakkeeran expresses his thanks to CSIR, Government of India, for providing him a Junior Research Fellowship.
REFERENCES 1. R. Sahadevan, K. M. Tamizhmani and M. Lakshamanan, J. Whys.A: Math. Cm. 19, 1783 (1985). 2. K. Porsezian, P. Shanmugha, K. Sundaram and A. Mahalingam, Phys. Rev. 50E, 1543 (1994). 3. K. Porsezian and T. Alagesan, Phys. Letf. A 198, 378 (1995). 4. M. J. Ablowitz, A. Ramani and H. Segur, J. Math. Phys. 21, 715 (1980). 5. Weiss, M. Tabor and G. Carnvale, J. Math. Phys. 24, 522 (1983). 6. Musette and R. Conte, J. Math. Phys. 32, 1450 (1991). 7. K. Porsezian and K. Nakkeeran, J. Mod. Opt. (In Press). 8. V. B. Matveev and M. A. Salle, Darboux Transformations and Solitons. Springer (1991). 9. K. Porsezian, J. Phys. A: Math. Gen 24, 337 (1991).
& Fractals Vol. 7, No. 3. pp. 377-.3X2. 1996 Copyright 0 1996 Elsevier Science Ltd Prmted in Great Britain. All rights reserved 096&-077Y/Y6 $15.00 + 0.00
0960-0779(95)00069-O
Singularity Structure Analysis and the Complete Integrability of the Higher Order Nonlinear Schriidinger-Maxwell-Bloch Equations* K. PORSEZIAN and K. NAKKEERAN Department of Physics, Anna University, Madras-600025, India (Accepted 4 July 1995)
propose the coupled system of the generalized nonlinear Schrodinger equation and the Maxwell-Bloch equations, which finds application in nonlinear fiber optics. Using the Painleve singularity structure analysis, the condition for the complete integrability is obtained. The Lax pair and other properties of the equations are discussed. With suitable parametric conditions, other interesting soliton models are generated.
Abstract-We
1. INTRODUCTION
Recently there has been considerable interest in the singularity structure analysis of the several physically interesting coupled nonlinear partial differential equations such as, the coupled nonlinear Schrodinger (NLS) equation [ 11, coupled higher order NLS equation [2], coupled Klein-Gordon equation [3] and so on. Nowadays the above analysis can also be used to investigate the coupled integrability properties of given partial differential equations like, the Lax pair, Backlund transformations, bilinear form, etc. Hence, the singularity structure analysis is considered to be one of the most powerful and systematic method in the field of nonlinear science. As large number of articles [4-61 are devoted to this analysis, we do not give a detailed introduction about the method. In this paper, we mainly consider one of the very important problems in the area of nonlinear fiber optics, namely the coupled higher order NLS-Maxwell-Bloch (HNLS-MB) equations in the form,
(1)
where E is the slowly varying envelope of electric field, the subscripts z and t denote the spatial and temporal partial derivatives, p and q are given by v1v2* and /v1I2 - Iv212 respectively. Here y1 and V~are the wave functions of the two energy levels of the resonant atoms. The bracketed term (. . .) means, the averaging over the entire frequency range, co (P(Z,
t; 0))
=
I --a
p(z,
cc
h(w)do= 1, I --a where h(o) is the uncertainty in the energy level. *Communicated by Professor Wadati. 371
W44
da
(2)
378
K. PORSEZIAN and K. NAKKEERAN
Equation (1) when E = 0 reduces to the NLS-MB equations [7],
E, = ~cI&~ + 1~12~1+ 24(p), pt = Ziwp + Eq, 7, = -(Ep*
(3)
+ Ep*),
when N = y = 0 and @= 1, (1) reduces to the derivative NLS-MB equations of the form, E, = ial[kE,
+ (E12E] - E@IE/~E, + 2a2(p),
pI = 2iwp + Ey, qr = -(Ep*
(4)
+ Ep*).
With CY= 0 and /I = y = 1, (1) results in mixed derivative NLS-MB equations which have the form, E, = icul[$,t + lE12E] - E[IEI~E, + E(lE12),1 + 2a~(p), pt = 2iwp + Ey, qt = -(Ep*
(5)
+ Ep”).
For @= 60~~and y = 0, coupled Hirota equation and Maxwell-Bloch (H-MB) can be derived from HNLS-MB equations. H-MB equations take the form, EZ = ial[iE,, pr = 2iop
+ jE12E] - E[E,, + 6jE(‘E,]
equations
+ 2cu,(p),
+ Er],
q, = -(Ep”
(6)
+ Ep*)
Thus it is understood that (1) is in general not integrable but contains several interesting families of equations, having very important applications in the study of the wave propagation in the erbium doped nonlinear fiber media. The physical application of the above papers will be published elsewhere. Except (3), the integrability aspects of the equations (1, 4-6), are still an open question. In order to investigate the integrability aspects of equation (1) we carry out a singularity structure analysis in Section 2, and show that (1) is integrable only for the condition 3a = b = 2y. After analyzing the Painleve property, the explicit Lax pair is also constructed. 2. PAINLEVk
ANALYSIS
To apply Painleve analysis, first one has to remove the averaging or integral function in (1). This can be solved by considering the function h(w) in (1) as a Dirac-delta function at the resonant frequency. Under this assumption the averaging function over the entire frequency range can be eliminated from the variable p. Further, for convenience, independent variables z and t are interchanged. With all these modifications, (1) takes the form, E, = ial[~E,, pz = 2iup rjz = -(Ep*
+ IEj2E] + Ev,
- &[cYYE~~~ + ,GIE12E, + yE(IEi2),]
+ ~CCY,P, (7)
+ Ep”).
The nature of the dependent variables E and p are complex, where as q is a real varip=c, p*=d and q=e able. To apply Painleve analysis, we express E = a, E*=b,
Integrability of the higher order nonlinear Schrbdinger-Maxwell-Bloch
equations
379
(* represents a complex conjugate). With the new dependent variables a, . . ., e, the set of (7) becomes a, = i&2,,
+ a2b) - E[auzzZ+ (/3 + y)aba, + yu2b,] + ~CQC,
b, = -k&b,,
+ ab2) - E[ab,,, + (p + y)abb, + yb’u,]
+ 2azd,
cz = 2ioc + 2ue, d, = -2iwd eZ = -(ad
(8)
+ 2be, + bc).
The generalized Laurant series expansions of a, . . . , e are, U =
W’CUj(ZY
t)q’(Z,
t)l
t)q’(Z>
‘)>
‘)V’(zy
‘)2
t)@(Z,
f).
j=O
@iCj(Z>
C =
(9)
j=O
d = q’Cdj(Z, j=O
e=
qqCej(Z, j=O
With uo, . . ., e. # 0; where 1, m, II, Y and q are negative integers; Uj, . . . , ej are the set of expansion coefficients which are analytic in the neighbourhood of the non-characteristic singular manifold QI(Z, t) = z + v(t) = 0. Looking at the leading order, we substitute u=uOcp’, . . ., e=eoqq in equation (8) and upon balancing terms we obtain the following results, I=
m = -1,
aobo = -1,
n=r=q
(10)
boco = aado.
Substituting the full expansion of the Laurent series and keeping the leading order terms alone, we obtain the following equation, A Bb; -2eo 0 do
Bu:, A 0 -2eo aa
0 (i -f n> 0 bo
0 0 (j +” 4 a0
0 0 -2uo -2bo (j + 4
= 0
(11)
where A = a(j - l)(j
- 2)(j - 3) + (j - l)(a + /3)uobo - (0 + 3cu)uobo,
B = (j - 1) - (p + y). Substituting (10) and solving the determinant the resonance values are found to be, j = -l,O,
3,4, -n, f
112 (174
380
K. PORSEZIAN and K. NAKKEERAN
From careful analysis, we find that (11) admits a sufficient number of positive resonances only when y1= -2 and for the condition 3a = p = 2y which emphasizes that (8) is non-integrable for other values of a, /3 and y. Substituting the values of n, N, /I?and y, the resonance values are found to be, j = -1, 0, 0, 2, 2, 3, 4, 4, 4.
(13) As usual, the resonance value at j = -1 corresponds to the arbitrariness of the singularity manifold 91and j = 0, 0 implies that a0 or b. and co or do are arbitrary which is evident from (10). Similarly from the coefficients of (qm3, qm3, CJ-‘, cp-*, qe2) we obtain,
(14) i EbO(P Ebo(B 2eo 40++ 37) Y)
Eao(p eao(8 2e0 co 0++ 3y) r>
bo 01
a0 01
2bo 2u, -10 i
Solving (14), the values of aI, . . ., el are found to be, al = 0, bl = 0, cl = 2iaoeo,
(15)
d, = -2iboeo, el = 0. Similarly we also obtain the matrix for a2, . . ., e2 as, ’ -2Ey
db:, 2eo 0
&/la: -2&Y 0 2eo
0 0 0 0
0 0 0 0
0 0 2a. 2bo
co
bo
QO
0
, do
a2 b2 c2 4 e2
F -2icr - 2a,e, 2idl - 2b,e, -aId, - b,c,
,
(16)
where D = -u~v,~~- ia,(2aoboal + aibl) - E/h.oa,b, - Ey(aoalb, + a:bo) - ~cY~c~~, F = -boqt + ial(2aobob, + bf,a,) - &fib,a,b, - &y(boalbl + b:ao) - 2a2do. Equation (16) reveals that there are two arbitrarinesses, corresponding to the resonances j = 2, 2. Also f rom the remaining powers of Q?,we find that (8) admits a sufficient number of arbitrary functions only for ~CV= p = 2y. Hence we conclude that the equations in (6) are expected to be integrable only for the above choice of parameters. So, the integrable version of (1) takes the form, E, = ial[$E,
+ /Ej*E] - &[Ettt + 3jEIZE, + $E(IEj*),] + 2a2(p),
pt = 2iop + Eq, qt = -(Ep*
(17)
+ Ep”).
Hence (17) admits the Painleve property and is expected to be integrable only for the above parametric choice. Then to establish the integrability properties, we construct the linear eigenvalue problem in the following form,
Integrability
of the higher order nonlinear
SchrGdinger-Maxwell-Bloch
equations
381
(18)
where
-i- &[(EE; - EXE*) + +]
+ 2icu,
0 -p"e" i,A- io 1
-r i A - -iBiw i pe 1,A- io 1
p*e” i I A- io.1 0
Here 0 = l/6&(2 - t/l&), x = z, - t/12~ and ?, (= A1+ i&) is the spectral parameter, so that the consistency condition Ur, - VI, + [ Ur, VI] = 0 leads to HNLS-MB equations. The construction of the soliton solution is in progress. 3. PAINLEVI?
ANALYSIS
OF THE HIROTA-MB
EQUATIONS
As we have discussed in the introduction, when /II = 6a and y = 0, (1) reduces to Hirota-MB (H-MB) equations. Substituting /3 = 6cx, y = 0 and n = -2 in (12), we get the resonances for the H-MB equations as j = -1, 0, 0, 1, 2, 3, 4, 4, 5. (19) Equation (6) also admits the sufficient number of arbitrary functions. The 2 X 2 linear eigenvalue problem of (6) is Yv, = U,Y, Yv, = V,Y, where
Y = (Y,
Y,)T
cw
K. PORSEZIAN and K. NAKKEERAN
382
Here G
= -4iE/i3
H = 4id’E
-
2ia,h2 + 2id\E)*
+ 2a,AE + 2idE,
+ ia,jE/* + E(EE; - E,E*), - 2&jE12E + ialE,
- EE,.
The H-MB equations are obtained from the consistency condition U,, - V,, + [V,, V,] = 0. Using Blcklund transformations [S], the single soliton solution [9] of the H-MB equation is obtained as E(e,t) = 2k2sech(p)exp(io
- if&),
(21)
where p and o are functions of z, t and soliton velocity parameters given by,
cT(z, t) = 2A,t + 4a,(A: - A:) -
m 2c~~(A~- w)h(o)dw I --cc A; + (A1 - o$
+ 8~Iz~(/tf
-
32;)
1
2 + do),
(p”’ and o(O)are independent of both z and t). And 6r is a real constant. 4. CONCLUSION
Thus, in this paper we have investigated the singularity structure analysis of the HNLS-MB equations which describes the wave propagation in erbium doped nonlinear fiber media. From the detailed analysis, we find the system admits Painleve property only for a particular condition between the parameters involved in the equations. We also discussed the other integrable systems involved in our proposed model. The Lax pair is explicitly constructed. Acknowledgements-K. Porsezian wishes to thank the Council of Scientific and Industrial Research and the Department of Science and Technology, Government of India, for their financial assistance through projects. The author K. Nakkeeran expresses his thanks to CSIR, Government of India, for providing him a Junior Research Fellowship.
REFERENCES 1. R. Sahadevan, K. M. Tamizhmani and M. Lakshamanan, J. Whys.A: Math. Cm. 19, 1783 (1985). 2. K. Porsezian, P. Shanmugha, K. Sundaram and A. Mahalingam, Phys. Rev. 50E, 1543 (1994). 3. K. Porsezian and T. Alagesan, Phys. Letf. A 198, 378 (1995). 4. M. J. Ablowitz, A. Ramani and H. Segur, J. Math. Phys. 21, 715 (1980). 5. Weiss, M. Tabor and G. Carnvale, J. Math. Phys. 24, 522 (1983). 6. Musette and R. Conte, J. Math. Phys. 32, 1450 (1991). 7. K. Porsezian and K. Nakkeeran, J. Mod. Opt. (In Press). 8. V. B. Matveev and M. A. Salle, Darboux Transformations and Solitons. Springer (1991). 9. K. Porsezian, J. Phys. A: Math. Gen 24, 337 (1991).