- Email: [email protected]
Black optical solitons for media with parabolic nonlinearity law in the presence of fourth order dispersion
15 May 2000
Optics Communications 178 Ž2000. 457–460 www.elsevier.comrlocateroptcom
Black optical solitons for media with parabolic nonlinearity law in the presence of fourth order dispersion S.L. Palacios ) , J.M. Fernandez-Dıaz ´ ´ Departamento de Fısica, UniÕersidad de OÕiedo, C r CalÕo Sotelo s r n, OÕiedo, E-33007, Spain ´ Received 27 January 2000; received in revised form 27 March 2000; accepted 29 March 2000
Abstract We find black solitary wave solutions for a nonlinear Schrodinger equation for media with a parabolic nonlinearity law in ¨ the presence of fourth-order dispersion. Particular cases are discussed. q 2000 Published by Elsevier Science B.V. All rights reserved. PACS: 42.65.y k; 42.65.Tg; 42.81.Dp Keywords: Dark solitons; Non-Kerr medium; Fourth-order dispersion
It is well known that the nonlinear Schrodinger ¨ equation ŽNLSE. A z s yi
b2 2
A t t q ig < A < 2A
Ž 1.
describes optical pulse propagation in optical fibers Žnonlinear media of the Kerr type. when the pulsewidth is above 100 fs. Such equation predicts bright soliton solutions in the anomalous dispersion region of the fiber while in the normal dispersion regime, bright pulses cannot propagate as solitons and the interplay between the group velocity dispersion ŽGVD. and the nonlinear index of refraction leads to spectral and temporal broadening of the
) Corresponding author. Tel.: q34-985-102-879; fax: q34-985103-324; e-mail: [email protected]
propagating pulses. Although bright solitons are allowed only for negative GVD, the NLSE also admits other soliton solutions for positive GVD. These solutions are called ‘dark pulse solitons’ Žthe intensity profile contains a dip in a uniform background. w1–3x. There exist several essential differences between bright and dark solitons. One of them consists of the existence of multiple bound states that can form bright solitons in clear contrast with dark solitons. In addition, given a fixed optical frequency and background intensity, there is a continuous range of dark solitons with different blackness parameters Žthe so called grey, black and darker than black solitons.. On the other hand, for a fixed frequency and intensity, just one bright soliton solution is possible. Finally, dark solitons have a phase profile which is an anti-symmetric function of time, whereas bright solitons have a constant phase. It is this phase func-
0030-4018r00r$ - see front matter q 2000 Published by Elsevier Science B.V. All rights reserved. PII: S 0 0 3 0 - 4 0 1 8 Ž 0 0 . 0 0 6 6 6 - 0
S.L. Palacios, J.M. Fernandez-Dıazr ´ ´ Optics Communications 178 (2000) 457–460
458
tion that is a major reason for the difficulty in generating and studying dark solitons experimentally. Since the early work of Zakharov and Shabat w4x and Hasegawa and Tappert w5,6x, dark optical solitons have been an active topic of research both theoretically and experimentally Žsee, for instance, the review w7x and the references therein.. One main reason for such a pursuit could be the possible application of dark solitons for long-distance optical communications taking advantage of its stability under the influence of the material losses, a lower Gordon–Haus jitter w8,9x, etc. Indeed, recently, dark solitons have been used to carry a 10 Gbitrs signal over 1200 km w10x. The stability cited above has already been demonstrated for the NLSE dark solitons when the material nonlinearity is of the Kerr type. However, when the nonlinear refractive index of an optical medium is illuminated with large light intensities the nonlinearity term in the NLSE differs from that of the Kerr law and the resulting equation is A z s yi
b2 2
Att q f Ž < A< 2 . A
Ž 2.
where the arbitrary function f describes a more general form of the intensity dependent refractive index and several different models Žpower-law, parabolic, dual power-law, saturable, exponential and others. have been proposed w3,11,12x. Although stationary pulses exist and some solutions can be written in analytic form, their behavior is essentially different from that of solutions of the NLSE w13–15x. This means that the pulse solutions of Ž2. are not solitons in the mathematical sense Žstrictly speaking they are solitary waves.. However, the use of the word ‘soliton’ has become common in the literature. We will sometimes use this term in the rest of the paper, but always taking into account this distinction. Thus far, only second order dispersion has been considered in the wave equations describing nonlinear propagation of optical pulses. However, if short pulses have to be injected Žto nearly 50 fs. third order dispersion becomes important and must be included in the equation w1–3x. Moreover, as the pulses become extremely short Žbelow 10 fs. fourth order dispersion also must be
taken into account w16–19x. Although some authors w20–22x have found bright optical solitons for a NLSE in the presence of fourth order dispersion Žthird order dispersion being nil., the dark soliton solutions are still unknown. In this paper we consider a higher order nonlinear Schrodinger equation including fourth order disper¨ sion with a parabolic nonlinearity law. This high dispersive cubic-quintic nonlinear Schrodinger equa¨ tion can be written in the form
A z s yi
b2 2
Att q
q ig 2 < A < 4A
b3 6
Attt q i
b4 24
A t t t t q ig 1 < A < 2A
Ž 3.
where b 2 is the parameter of the group velocity dispersion ŽGVD., b 3 and b4 are the third order and fourth order dispersions, respectively, and g 1 and g 2 are the nonlinearity coefficients. When the higher order terms are ignored we obtain the NLSE, which is inverse scattering transform ŽIST. integrable. This means that it is possible to find both solitary wave and multi-soliton solutions. Searching for multi-soliton solutions always implies solving an initial value problem. This can be a complicated task and, in fact, is not always possible. On the other hand, the integrability of a nonlinear equation can be studied applying the so called Painleve´ analysis. It is widely believed that possession of the Painleve´ property is a sufficient criterion for integrability. It is remarkable that non-integrability is not necessarily related to the nonlinear terms. Higher order dispersions, for example, also can make the system to be non-integrable Žwhile it remains Hamiltonian.. Finally, there exists another approach to find exact solutions of nonlinear evolution equations. This technique basically consists of expressing the solution in terms of an amplitude and a phase function. The formalism is quite simple but it presents an important drawback: it only permits us to obtain solitary wave solutions. We will make use of this formalism looking for a solution with an amplitude that only depends on the
S.L. Palacios, J.M. Fernandez-Dıazr ´ ´ Optics Communications 178 (2000) 457–460
time and a phase only depending on the coordinate in the direction of propagation such as A Ž z ,t . s A 0 tanh Ž trt 0 . e i k z where A 0 , t 0 and k are the unknown parameters representing the amplitude, pulsewidth and wavevector per unit length, respectively, to be determined. Substituting in Ž3. and equating the coefficients corresponding to identical powers of ‘tanh’ we obtain the following system of three equations Žafter separating the real and imaginary terms.
b2
q
t 02 y
b2 t 02
2 b4 3 t 04
y
sk
3
q g 1 A20 s 0 t 04
The equation corresponding to the real part implies that the third order dispersion is identically zero Žphysically, this implies a special choice of the carrier frequency.. Solving for the above system and after some simple algebraic manipulations we conclude that
t0 s
ks
)
15
y
5
'15 3
)
g2
In this case, the nonlinear indices of refraction of the medium are of the focusing type. It is obvious that the fourth-order dispersion coefficient must be negative Ž b4 - 0. and the GVD must be positive Ž b 2 ) 0. with the following restriction:
ž
)
g 12 g2
in order to obtain positive values of the amplitude, A 0 , and the pulsewidth, t 0 .
q g 2 A40 s 0. t 04
1
1. Two nonlinearities positive ( g 1 ,g 2 ) 0)
yb4
b4
A0 s
Next, we analyze the particular cases that arise. It is obvious from the above expressions that in all cases the b4 coefficient and the nonlinearity coefficient g 2 must have opposite signs Žg 2 b4 - 0..
b 22
5 b4
459
g1 y
b2
(y b rg 4
2
/
b4
2. Two nonlinearities negative ( g 1 ,g 2 - 0) Now, the nonlinear medium is characterized by two defocusing nonlinear indices of refraction. For the optical pulse parameters to be physically meaningful the b4 coefficient must be positive Ž b4 ) 0.. Proceeding in a way analogous to the previous case we obtain the following conditions
b 2 - 0;
(
yb 2 q g 1 y b4rg 2
3
Ž y b 2 qg 1'y b4 rg 2 . Ž 3 b 2 q2g 1'y b4 rg 2 .
25
b4
.
From these values of the pulse parameters, it is simple to see that both the amplitude and the width of the soliton are uniquely determined from the characteristics of the nonlinear medium, i.e. the second and fourth order dispersion coefficients and the two nonlinear coefficients. This notable feature make these solitons essentially different from those of the NLSE type where a continuous set of pulse amplitude and width exists.
b 22 b4
)
g 12 yg 2
with positive values of the GVD Ž b 2 ) 0. not being permitted. The results obtained in the two previous paragraphs can be summarized if we define a dispersion ratio coefficient ŽDRC., R D , and a nonlinearity ratio coefficient ŽNRC., R N in the forms RD s
RN s
b 22 < b4 <
g 12
.
S.L. Palacios, J.M. Fernandez-Dıazr ´ ´ Optics Communications 178 (2000) 457–460
460
So, our conclusions for the existence of black solitons when both nonlinearities are of the same sign can be expressed by the following condition RD ) RN .
3. Negative high power nonlinearity ( g 1 ) 0, g 2 - 0) Now the curvature of the group velocity dispersion must be positive Ž b4 ) 0.. This condition implies that b2 - 0 or
b 2 ) 0;
b 22 b4
-
g 12 yg 2
.
Therefore, the propagation of dark solitary waves is possible when the second order dispersion is either positive or negative. This fact is completely different from that of the high dispersive cubic nonlinear Schrodinger equation Žour equation with just the ¨ cubic nonlinearity. in which the GVD coefficient and b4 must be both negative. This kind of nonlinearity Žfocusing for weak light intensities and defocusing for large light intensities. has been found recently in organic materials such as polydiacetylene paratoluene sulfonate ŽPTS. w23x. 4. Positive high power nonlinearity ( g 1 - 0, g 2 ) 0) Analogously, we conclude that b4 - 0 and b2 ) 0 or
b 2 - 0;
b 22 yb4
)
g 12 g2
In summary, we have obtained a dark optical solitary wave solution for a high dispersive Žfourthorder. nonlinear Schrodinger equation for nonlinear ¨ media described by two nonlinearities Žcubic and quintic.. Particular cases concerning the focusing or defocusing nature of the nonlinear indices of refraction have been analyzed. These pulses can only occur for just one value of its pulsewidth, amplitude and phase shift per unit length. In addition, when the two nonlinearities are of the same sign forbidden regions for the GVD appear Žespecially, that corresponding to opposite signs of b 2 and b4 .. On the other side, when the nonlinearities are competitive, i.e., g 1 g 2 - 0 it appears the possibility of the existence of dark solitary waves for both positive and negative values of GVD.
.
The same conclusions as in the previous case concerning the two possible signs of b 2 apply. Again, the results obtained for the saturating case Žboth nonlinearities opposite in sign. can be summarized in terms of the DRC and the NRC. The condition for black solitons to exist is RD - RN .
References w1x G.P. Agrawal, Nonlinear Fiber Optics, Academic Press, San Diego, 1995. w2x A. Hasegawa, Y. Kodama, Solitons in Optical Communications, Oxford University Press, New York, 1995. w3x N.N. Akhmediev, A. Ankiewicz, Solitons: Nonlinear Pulses and Beams, Chapman and Hall, London, 1997. w4x V.E. Zakharov, A.B. Shabat, Soviet Phys. JETP 34 Ž1972. 62. w5x A. Hasegawa, F.D. Tappert, Appl. Phys. Lett. 23 Ž1973. 142. w6x A. Hasegawa, F.D. Tappert, Appl. Phys. Lett. 23 Ž1973. 171. w7x Yu.S. Kivshar, B. Luther-Davies, Phys. Rep. 298 Ž1998. 81. w8x Y. Chen, J. Atai, IEEE J. Quantum Electron. 34 Ž1998. 1301. w9x Yu.S. Kivshar, M. Haelterman, Ph. Emplit, J.P. Hamaide, Opt. Lett. 19 Ž1994. 19. w10x M. Nakazawa, K. Suzuki, Electron. Lett. 31 Ž1995. 1076. w11x D.E. Pelinovsky, Yu.S. Kivshar, V.V. Afanasjev, Phys. Rev. E 54 Ž1996. 2015. w12x J. Jasinski, Opt. Commun. 172 Ž1999. 325. ´ w13x K. Hayata, M. Koshiba, Phys. Rev. E 51 Ž1995. 1499. w14x R.W. Micallef, V.V. Afanasjev, Yu.S. Kivshar, J.D. Love, Phys. Rev. E 54 Ž1996. 2936. ¨ w15x K. Dimitrevski, E. Reimhult, E. Svensson, A. Ohgren, D. Anderson, A. Berntson, M. Lisak, M.L. Quiroga-Teixeiro, Phys. Lett. A 248 Ž1998. 369. w16x I.P. Christov, M.M. Murnane, H.C. Kapteyn, J. Zhou, C.P. Huang, Opt. Lett. 19 Ž1994. 1465. w17x V.I. Karpman, A.G. Shagalov, Phys. Lett. A 228 Ž1997. 59. w18x V.I. Karpman, Phys. Lett. A 244 Ž1998. 397. w19x A.G. Karpman, V.I. Shagalov, Phys. Lett. A 254 Ž1999. 319. w20x A. Hook, ¨¨ M. Karlsson, Opt. Lett. 18 Ž1993. 1388. w21x M. Karlsson, A. Hook, ¨¨ Opt. Commun. 104 Ž1994. 303. w22x M. Piche, ´ J.F. Cormier, X. Zhu, Opt. Lett. 21 Ž1996. 845. w23x B.L. Lawrence, G.I. Stegeman, Opt. Lett. 23 Ž1998. 591.
Optics Communications 178 Ž2000. 457–460 www.elsevier.comrlocateroptcom
Black optical solitons for media with parabolic nonlinearity law in the presence of fourth order dispersion S.L. Palacios ) , J.M. Fernandez-Dıaz ´ ´ Departamento de Fısica, UniÕersidad de OÕiedo, C r CalÕo Sotelo s r n, OÕiedo, E-33007, Spain ´ Received 27 January 2000; received in revised form 27 March 2000; accepted 29 March 2000
Abstract We find black solitary wave solutions for a nonlinear Schrodinger equation for media with a parabolic nonlinearity law in ¨ the presence of fourth-order dispersion. Particular cases are discussed. q 2000 Published by Elsevier Science B.V. All rights reserved. PACS: 42.65.y k; 42.65.Tg; 42.81.Dp Keywords: Dark solitons; Non-Kerr medium; Fourth-order dispersion
It is well known that the nonlinear Schrodinger ¨ equation ŽNLSE. A z s yi
b2 2
A t t q ig < A < 2A
Ž 1.
describes optical pulse propagation in optical fibers Žnonlinear media of the Kerr type. when the pulsewidth is above 100 fs. Such equation predicts bright soliton solutions in the anomalous dispersion region of the fiber while in the normal dispersion regime, bright pulses cannot propagate as solitons and the interplay between the group velocity dispersion ŽGVD. and the nonlinear index of refraction leads to spectral and temporal broadening of the
) Corresponding author. Tel.: q34-985-102-879; fax: q34-985103-324; e-mail: [email protected]
propagating pulses. Although bright solitons are allowed only for negative GVD, the NLSE also admits other soliton solutions for positive GVD. These solutions are called ‘dark pulse solitons’ Žthe intensity profile contains a dip in a uniform background. w1–3x. There exist several essential differences between bright and dark solitons. One of them consists of the existence of multiple bound states that can form bright solitons in clear contrast with dark solitons. In addition, given a fixed optical frequency and background intensity, there is a continuous range of dark solitons with different blackness parameters Žthe so called grey, black and darker than black solitons.. On the other hand, for a fixed frequency and intensity, just one bright soliton solution is possible. Finally, dark solitons have a phase profile which is an anti-symmetric function of time, whereas bright solitons have a constant phase. It is this phase func-
0030-4018r00r$ - see front matter q 2000 Published by Elsevier Science B.V. All rights reserved. PII: S 0 0 3 0 - 4 0 1 8 Ž 0 0 . 0 0 6 6 6 - 0
S.L. Palacios, J.M. Fernandez-Dıazr ´ ´ Optics Communications 178 (2000) 457–460
458
tion that is a major reason for the difficulty in generating and studying dark solitons experimentally. Since the early work of Zakharov and Shabat w4x and Hasegawa and Tappert w5,6x, dark optical solitons have been an active topic of research both theoretically and experimentally Žsee, for instance, the review w7x and the references therein.. One main reason for such a pursuit could be the possible application of dark solitons for long-distance optical communications taking advantage of its stability under the influence of the material losses, a lower Gordon–Haus jitter w8,9x, etc. Indeed, recently, dark solitons have been used to carry a 10 Gbitrs signal over 1200 km w10x. The stability cited above has already been demonstrated for the NLSE dark solitons when the material nonlinearity is of the Kerr type. However, when the nonlinear refractive index of an optical medium is illuminated with large light intensities the nonlinearity term in the NLSE differs from that of the Kerr law and the resulting equation is A z s yi
b2 2
Att q f Ž < A< 2 . A
Ž 2.
where the arbitrary function f describes a more general form of the intensity dependent refractive index and several different models Žpower-law, parabolic, dual power-law, saturable, exponential and others. have been proposed w3,11,12x. Although stationary pulses exist and some solutions can be written in analytic form, their behavior is essentially different from that of solutions of the NLSE w13–15x. This means that the pulse solutions of Ž2. are not solitons in the mathematical sense Žstrictly speaking they are solitary waves.. However, the use of the word ‘soliton’ has become common in the literature. We will sometimes use this term in the rest of the paper, but always taking into account this distinction. Thus far, only second order dispersion has been considered in the wave equations describing nonlinear propagation of optical pulses. However, if short pulses have to be injected Žto nearly 50 fs. third order dispersion becomes important and must be included in the equation w1–3x. Moreover, as the pulses become extremely short Žbelow 10 fs. fourth order dispersion also must be
taken into account w16–19x. Although some authors w20–22x have found bright optical solitons for a NLSE in the presence of fourth order dispersion Žthird order dispersion being nil., the dark soliton solutions are still unknown. In this paper we consider a higher order nonlinear Schrodinger equation including fourth order disper¨ sion with a parabolic nonlinearity law. This high dispersive cubic-quintic nonlinear Schrodinger equa¨ tion can be written in the form
A z s yi
b2 2
Att q
q ig 2 < A < 4A
b3 6
Attt q i
b4 24
A t t t t q ig 1 < A < 2A
Ž 3.
where b 2 is the parameter of the group velocity dispersion ŽGVD., b 3 and b4 are the third order and fourth order dispersions, respectively, and g 1 and g 2 are the nonlinearity coefficients. When the higher order terms are ignored we obtain the NLSE, which is inverse scattering transform ŽIST. integrable. This means that it is possible to find both solitary wave and multi-soliton solutions. Searching for multi-soliton solutions always implies solving an initial value problem. This can be a complicated task and, in fact, is not always possible. On the other hand, the integrability of a nonlinear equation can be studied applying the so called Painleve´ analysis. It is widely believed that possession of the Painleve´ property is a sufficient criterion for integrability. It is remarkable that non-integrability is not necessarily related to the nonlinear terms. Higher order dispersions, for example, also can make the system to be non-integrable Žwhile it remains Hamiltonian.. Finally, there exists another approach to find exact solutions of nonlinear evolution equations. This technique basically consists of expressing the solution in terms of an amplitude and a phase function. The formalism is quite simple but it presents an important drawback: it only permits us to obtain solitary wave solutions. We will make use of this formalism looking for a solution with an amplitude that only depends on the
S.L. Palacios, J.M. Fernandez-Dıazr ´ ´ Optics Communications 178 (2000) 457–460
time and a phase only depending on the coordinate in the direction of propagation such as A Ž z ,t . s A 0 tanh Ž trt 0 . e i k z where A 0 , t 0 and k are the unknown parameters representing the amplitude, pulsewidth and wavevector per unit length, respectively, to be determined. Substituting in Ž3. and equating the coefficients corresponding to identical powers of ‘tanh’ we obtain the following system of three equations Žafter separating the real and imaginary terms.
b2
q
t 02 y
b2 t 02
2 b4 3 t 04
y
sk
3
q g 1 A20 s 0 t 04
The equation corresponding to the real part implies that the third order dispersion is identically zero Žphysically, this implies a special choice of the carrier frequency.. Solving for the above system and after some simple algebraic manipulations we conclude that
t0 s
ks
)
15
y
5
'15 3
)
g2
In this case, the nonlinear indices of refraction of the medium are of the focusing type. It is obvious that the fourth-order dispersion coefficient must be negative Ž b4 - 0. and the GVD must be positive Ž b 2 ) 0. with the following restriction:
ž
)
g 12 g2
in order to obtain positive values of the amplitude, A 0 , and the pulsewidth, t 0 .
q g 2 A40 s 0. t 04
1
1. Two nonlinearities positive ( g 1 ,g 2 ) 0)
yb4
b4
A0 s
Next, we analyze the particular cases that arise. It is obvious from the above expressions that in all cases the b4 coefficient and the nonlinearity coefficient g 2 must have opposite signs Žg 2 b4 - 0..
b 22
5 b4
459
g1 y
b2
(y b rg 4
2
/
b4
2. Two nonlinearities negative ( g 1 ,g 2 - 0) Now, the nonlinear medium is characterized by two defocusing nonlinear indices of refraction. For the optical pulse parameters to be physically meaningful the b4 coefficient must be positive Ž b4 ) 0.. Proceeding in a way analogous to the previous case we obtain the following conditions
b 2 - 0;
(
yb 2 q g 1 y b4rg 2
3
Ž y b 2 qg 1'y b4 rg 2 . Ž 3 b 2 q2g 1'y b4 rg 2 .
25
b4
.
From these values of the pulse parameters, it is simple to see that both the amplitude and the width of the soliton are uniquely determined from the characteristics of the nonlinear medium, i.e. the second and fourth order dispersion coefficients and the two nonlinear coefficients. This notable feature make these solitons essentially different from those of the NLSE type where a continuous set of pulse amplitude and width exists.
b 22 b4
)
g 12 yg 2
with positive values of the GVD Ž b 2 ) 0. not being permitted. The results obtained in the two previous paragraphs can be summarized if we define a dispersion ratio coefficient ŽDRC., R D , and a nonlinearity ratio coefficient ŽNRC., R N in the forms RD s
RN s
b 22 < b4 <
g 12
.
S.L. Palacios, J.M. Fernandez-Dıazr ´ ´ Optics Communications 178 (2000) 457–460
460
So, our conclusions for the existence of black solitons when both nonlinearities are of the same sign can be expressed by the following condition RD ) RN .
3. Negative high power nonlinearity ( g 1 ) 0, g 2 - 0) Now the curvature of the group velocity dispersion must be positive Ž b4 ) 0.. This condition implies that b2 - 0 or
b 2 ) 0;
b 22 b4
-
g 12 yg 2
.
Therefore, the propagation of dark solitary waves is possible when the second order dispersion is either positive or negative. This fact is completely different from that of the high dispersive cubic nonlinear Schrodinger equation Žour equation with just the ¨ cubic nonlinearity. in which the GVD coefficient and b4 must be both negative. This kind of nonlinearity Žfocusing for weak light intensities and defocusing for large light intensities. has been found recently in organic materials such as polydiacetylene paratoluene sulfonate ŽPTS. w23x. 4. Positive high power nonlinearity ( g 1 - 0, g 2 ) 0) Analogously, we conclude that b4 - 0 and b2 ) 0 or
b 2 - 0;
b 22 yb4
)
g 12 g2
In summary, we have obtained a dark optical solitary wave solution for a high dispersive Žfourthorder. nonlinear Schrodinger equation for nonlinear ¨ media described by two nonlinearities Žcubic and quintic.. Particular cases concerning the focusing or defocusing nature of the nonlinear indices of refraction have been analyzed. These pulses can only occur for just one value of its pulsewidth, amplitude and phase shift per unit length. In addition, when the two nonlinearities are of the same sign forbidden regions for the GVD appear Žespecially, that corresponding to opposite signs of b 2 and b4 .. On the other side, when the nonlinearities are competitive, i.e., g 1 g 2 - 0 it appears the possibility of the existence of dark solitary waves for both positive and negative values of GVD.
.
The same conclusions as in the previous case concerning the two possible signs of b 2 apply. Again, the results obtained for the saturating case Žboth nonlinearities opposite in sign. can be summarized in terms of the DRC and the NRC. The condition for black solitons to exist is RD - RN .
References w1x G.P. Agrawal, Nonlinear Fiber Optics, Academic Press, San Diego, 1995. w2x A. Hasegawa, Y. Kodama, Solitons in Optical Communications, Oxford University Press, New York, 1995. w3x N.N. Akhmediev, A. Ankiewicz, Solitons: Nonlinear Pulses and Beams, Chapman and Hall, London, 1997. w4x V.E. Zakharov, A.B. Shabat, Soviet Phys. JETP 34 Ž1972. 62. w5x A. Hasegawa, F.D. Tappert, Appl. Phys. Lett. 23 Ž1973. 142. w6x A. Hasegawa, F.D. Tappert, Appl. Phys. Lett. 23 Ž1973. 171. w7x Yu.S. Kivshar, B. Luther-Davies, Phys. Rep. 298 Ž1998. 81. w8x Y. Chen, J. Atai, IEEE J. Quantum Electron. 34 Ž1998. 1301. w9x Yu.S. Kivshar, M. Haelterman, Ph. Emplit, J.P. Hamaide, Opt. Lett. 19 Ž1994. 19. w10x M. Nakazawa, K. Suzuki, Electron. Lett. 31 Ž1995. 1076. w11x D.E. Pelinovsky, Yu.S. Kivshar, V.V. Afanasjev, Phys. Rev. E 54 Ž1996. 2015. w12x J. Jasinski, Opt. Commun. 172 Ž1999. 325. ´ w13x K. Hayata, M. Koshiba, Phys. Rev. E 51 Ž1995. 1499. w14x R.W. Micallef, V.V. Afanasjev, Yu.S. Kivshar, J.D. Love, Phys. Rev. E 54 Ž1996. 2936. ¨ w15x K. Dimitrevski, E. Reimhult, E. Svensson, A. Ohgren, D. Anderson, A. Berntson, M. Lisak, M.L. Quiroga-Teixeiro, Phys. Lett. A 248 Ž1998. 369. w16x I.P. Christov, M.M. Murnane, H.C. Kapteyn, J. Zhou, C.P. Huang, Opt. Lett. 19 Ž1994. 1465. w17x V.I. Karpman, A.G. Shagalov, Phys. Lett. A 228 Ž1997. 59. w18x V.I. Karpman, Phys. Lett. A 244 Ž1998. 397. w19x A.G. Karpman, V.I. Shagalov, Phys. Lett. A 254 Ž1999. 319. w20x A. Hook, ¨¨ M. Karlsson, Opt. Lett. 18 Ž1993. 1388. w21x M. Karlsson, A. Hook, ¨¨ Opt. Commun. 104 Ž1994. 303. w22x M. Piche, ´ J.F. Cormier, X. Zhu, Opt. Lett. 21 Ž1996. 845. w23x B.L. Lawrence, G.I. Stegeman, Opt. Lett. 23 Ž1998. 591.