Non-integrability of equations governing pulse propagation in dispersion-managed optical fibers

6 September 1999

Physics Letters A 260 Ž1999. 68–77 www.elsevier.nlrlocaterphysleta

Non-integrability of equations governing pulse propagation in dispersion-managed optical fibers T.I. Lakoba 1 Rochester Theory Center for Optical Science & Engineering, Institute of Optics, P.O.Box 270186, UniÕersity of Rochester, Rochester, NY 14627, USA Received 21 April 1999; received in revised form 6 July 1999; accepted 26 July 1999 Communicated by A.P. Fordy

Abstract We show that the equation governing pulse propagation in dispersion-managed optical fibers, as well as the reduced form of that equation, does not have conserved or periodically conserved quantities other than the mass, momentum, and Žfor the reduced equation only. the Hamiltonian. Implications of this result for the problem of four-wave mixing in collisions of pulses in optical telecommunication channels, are discussed. q 1999 Published by Elsevier Science B.V. All rights reserved. PACS: 03.40.Kf; 42.65.Tg; 42.81.Dp Keywords: Integrability; Conservation laws; Dispersion management; Optical solitons; Four-wave mixing

1. Introduction The technique of dispersion management ŽDM., where one periodically compensates fiber dispersion by inserting in the transmission line segments of fiber Žor Bragg gratings. with the opposite sign of dispersion, has become widely used in optical telecommunications. For a review of advantages which DM offers for the soliton data transmission format, see, e.g., Refs. w1,2x. The equation governing pulse propagation in dispersion-managed ŽDM. fibers is the nonlinear Schrodinger equation ŽNLS. where ¨ the dispersion coefficient is a piecewise-constant, periodic function of the propagation distance. This

1

E-mail: [email protected]

equation can be written in the following non-dimensional form: i

Eu Ez

1 q 2

Ž d0 q D Ž z . .

E 2u Et 2

q u< u< 2 s 0 .

Ž 1.

Here we have used the fiber-optics convention, whereby the propagation distance, z, plays the role of the evolution variable, and the retarded time, t , the role of the spatial variable. The dispersion coefficient in Eq. Ž1. is explicitly written as a sum of its average, constant part, d 0 , and the periodic part, DŽ z ., whose average over the dispersion map period vanishes. Specifically, DŽ z . takes on values D1 and D 2 over the fiber segments of respective lengths L1 and L2 , which compose the map, and D 1 L1 q D 2 L 2 s 0 .

0375-9601r99r$ - see front matter q 1999 Published by Elsevier Science B.V. All rights reserved. PII: S 0 3 7 5 - 9 6 0 1 Ž 9 9 . 0 0 5 0 5 - 8

Ž 2.

T.I. Lakobar Physics Letters A 260 (1999) 68–77

One can further normalize variables in Eq. Ž1. so as to have w3x the map period equal to unity, i.e. L1 q L 2 s 1 ,

Ž 3.

Note that in Eqs. Ž6. and Ž7., the lower limit in the integral H z DŽ zX . dzX is not important, and in what follows we set it to be 0. In Eq. Ž5. and below we use the following shorthand notations:

and also

`

< D 1 L1 < s < D 2 L 2 < s 1 .

Ž 4.

Without loss of generality, in what follows we take D 1 L1 s q1 .

Ž 4’ .

In Eq. Ž1., we also assumed that the fiber is lossless. From the applications standpoint, the most interesting is the case of the so-called strong DM, where both d 0 and < u < 2 are on the order of some small parameter e . In that case, it was found numerically Žsee, e.g., Ref. w4x. that Eq. Ž1. can have long-living, pulse-like solutions for either sign of d 0 , and even for d 0 s 0. Those solutions had a remarkably regular structure with oscillating tails, which was explained in Refs. w5,6x by representing the solution of Eq. Ž1. as a superposition of certain Hermite–Gaussian functions Žsee also Ref. w7x.. Moreover, stationary solutions of the following integro-differential equation, ia˙ Ž v . y

1 2

d0 v 2 aŽ v .

q d123 d 12,3 v a Ž v 1 . a Ž v 2 . a ) Ž v 3 . T123 v s 0 ,

H

Ž 5. to which Eq. Ž1. is reduced in the strong DM limit Ži.e. for d 0 ; < u < 2 ; e < 1., were also found w8x to exhibit this regular structure. Eq. Ž5. was first derived in Ref. w9x; it was later re-derived in Refs. w8,10x by different techniques. In this equation, the overdot denotes ErE z, and aŽ v , z . s

1 2p

exp

i 2

v2

z

X

H D Ž z . dz

X

`

Hy`dt exp wyi vt x u Ž t , z . ,

=

Ž 6.

T123 v ' T Ž v 1 , v 2 , v 3 , v . s

i

1

H0 dzexp y 2 Ž v z

=

X

H D Ž z . dz

X

.

2 2 2 2 1 qv2 yv3 yv

. Ž 7.

69

Hd

123 s

`

`

Hy`Hy`Hy`d v d v 1

2 dv3

d 12 ,3 v s d Ž v 1 q v 2 y v 3 y v . .

,

Ž 8.

Given the numerical discovery in Ref. w4x of the highly regular structure of the pulse-like solutions of Eq. Ž1., one can ask whether that equation is integrable, or in some sense close to being integrable, at least for some values of its parameters DŽ z . and d 0 . A straightforward calculation using the Painleve´ test answers the above question to the negative; the details are presented in Appendix A. However, one can still consider the following possibilities. First, even though Eq. Ž1. is non-integrable, yet the reduced equation, Eq. Ž5., can be integrable w11x. Second, integrability Žat least in the Ž1 q 1.dimensional case. usually implies the existence of infinitely many conserved quantities for the equation in question. Thus one can ask whether Eq. Ž1. has any ‘quasi-conserved’ quantities which would change periodically in z. Below we call such quantities ‘periodically conserved’. It is natural to assume that the period with which such quantities could change equals that of the dispersion map. In this work, we use the method developed by Zakharov and Schulman w12,13x Žsee also Ref. w14x for a comprehensive review. and answer the above two questions to the negative. The rest of this work is organized as follows. In Section 2 ŽSection 3. we prove that Eq. Ž5. ŽEq. Ž1.. with d 0 / 0 can have no conserved Žperiodically conserved. quantities other than the mass, momentum, and the Hamiltonian. In fact, we show that for Eq. Ž1., even the Hamiltonian is not periodically conserved. However, it should be noted that Eq. Ž5. admits infinitely many functionals with a quadratic leading-order part, which are conserved up to the fourth order inclusively. A similar statement also holds for Eq. Ž1., with ‘conserved’ being replaced by ‘periodically conserved’, and provided that a certain condition on the coefficients DŽ z . and d 0 is satisfied. In Sections 2 and 3 we also comment on the case d 0 s 0, which requires a slightly different anal-

T.I. Lakobar Physics Letters A 260 (1999) 68–77

70

ysis. In that case, we were also unable to find any additional conserved Žor periodically conserved. quantities. Finally, in Section 4, we discuss the implications of our results for both the non-returnto-zero and soliton data transmission formats in optical telecommunications.

2. Conserved quantities of Eq. (5)

Ž4. I s d v f Ž v . < av < 2 q Ž d d a . 12 ,34 I1234

H

Ž6. q Ž d d a . 123,456 I123456 q PPP ,

Ž 9.

H

Ž4. where av s aŽ v ., a1 s aŽ v 1 ., I1234 s I Ž4. Ž v 1 , v 2 , ) v 3 , v4 ., Ž d d a.12,34 s d1234 d 12,34 a1 a2 a3 a4) , etc. We also note that due to our adoption of the fiber-optics notations in Eq. Ž1., the frequencies v play the same role as the wave vectors k played in Ref. w13x and related works. The functions I Ž4. and I Ž6. satisfy the following symmetry relations with respect to their arguments: Ž4. Ž4. Ž4. Ž4. ) I1234 s I2134 s I1243 s I3412 ,

Ž 10 .

Ž6. Ž6. ) I123456 s IPŽ6.Ž123. P Ž456. s I456123 ,

Ž 11 .

where P Ž123. stands for any permutation of 1,2,3. Note that T1234 , defined in Eq. Ž7., also satisfies conditions Ž10.. The requirement that I be conserved means that dIrdz s 0 at each successive order O Ž a n .. At the order O Ž a 2 ., this condition always holds. At the next order, O Ž a4 ., Eqs. Ž9. and Ž5. are used to obtain the condition

H Ž d d a.

12 ,34

ž

1 2

Ž4. Ž4. f 12,34 T1234 y d 0 I1234 D12,34 s0 ,

/

Ž 12 . where we have denoted f 12 ,34 s f Ž v 1 . q f Ž v 2 . y f Ž v 3 . y f Ž v4 . , Ž4. D12 ,34 s

1 2

Ž v 12 q v 22 y v 32 y v42 . .

Then the I Ž4.-term in expansion Ž9. is nonsingular provided that f 12,34 T1234 s 0 on the so-called w12x singular manifold

v 1 q v 2 y v 3 y v4 s 0,

Following Ref. w13x, we look for conserved quantities, I, of Eq. Ž5., in the following form:

H

In deriving Eq. Ž12., we have used symmetry conditions Ž10. for T1234 . Eq. Ž12. requires that f 12 ,34 Ž4. I1234 s T1234 . Ž 15 . Ž4. 2 d 0 D12,34

Ž 13 . Ž 14 .

Ž4. D12,34 s0 .

Ž 16 .

It is known w15x that Eq. Ž16. has only two solutions:

Ž v 1 s v 3 , v 2 s v4 . and Ž v 2 s v 3 , v 1 s v4 . ,

Ž 17 .

both of which imply f 12,34 s 0. Hence a nonsingular fourth-order term in expansion Ž9. can always be found, and therefore we need to consider the next order, O Ž a6 ., of dIrdz. There we find a condition

H Ž d d a.

123,456

Ž2

. I ŽŽ 445y3 . 345 T126 Ž 12y6 .

Ž4. Ž6. Ž6. Ž 12y6 . T Ž 45y3 . 345 y d 0 D123,456 I123456 . s 0 , yI126

Ž 18 . Ž6. Ž4. where D123,456 is defined similarly to D12,34 in Eq. Ž4. Ž4. Ž14., and IŽ45y3.345 s I Ž v4 q v 5 y v 3 , v 3 , v 4 , v 5 ., etc. From Eq. Ž18. we find, for d 0 / 0, that a nonsingular I Ž6. could only exist if one has

H Ž d d a.

123,456

. Ž IŽŽ 445y3 . 345 T126 Ž 12y6 .

Ž4. Ž 12y6 . T Ž 45y3 . 345 . s 0 yI126

Ž 19 .

on the singular manifold

v 1 q v 2 q v 3 y v4 y v 5 y v6 s 0,

Ž6. D123,456 s0 . Ž 20 .

We now show that Eqs. Ž19. and Ž20. cannot have a common solution, unless f Ž v . s A v 2 q Bv q C, where A, B,C are arbitrary constants. For the latter choice of f Ž v ., the conserved quantities are just the mass Žalso called the number of photons., momentum, and the Hamiltonian w12x. Ž4. First, on manifold Ž20., one has DŽ45y3.345 q Ž4. Ž D126Ž12y6. s 0. Then one can use Eq. 15. to transform the l.h.s. of Eq. Ž19. into the form TŽ45y3.345T126Ž12y6. f 123,456 , Ž d d a . 123,456 Ž 21 . Ž4. D12 ,6Ž12y6.

H

T.I. Lakobar Physics Letters A 260 (1999) 68–77

where f 123,456 s f Ž v 1 . q f Ž v 2 . q f Ž v 3 . y f Ž v4 . y f Ž v 5 . y f Ž v6 .. Next, it was shown in Ref. w15x that f 123,456 / 0 Žunless f Ž v . s A v 2 q Bv q C . on manifold Ž20., except on any of its submanifolds of the form Ž16.. Then Eq. Ž19. could only hold if the first factor of the integrand in expression Ž21., properly symmetrized, vanishes on the manifold Ž20.: Ti jnŽ i jyn.TŽ l myk . k l m s0 . Ž 22 . Ý DiŽ4.j, nŽ i jyn. ijksP Ž123 . lmnsP Ž456 .

In Eq. Ž22., indices i, j,k Ž l,m,n. take on values 1,2,3 Ž4,5,6. in any permutation; thus the sum on the l.h.s. contains 9 terms. To evaluate the l.h.s. of Eq. Ž22., we use the explicit expression for T1234 : T1234 s

Ž4. exp yi D12 ,34 y 1 Ž4. Ž yi . D12 ,34

,

Ž 23 .

found from Eqs. Ž7. and Ž2. – Ž4X ., and also the following parametrization of manifold Ž20. w15x: 1 1 v 1 s P q R u q y q 3Õ , Ž 24a. u Õ 1 1 v 2 s P q R u q q y 3Õ , Ž 24b. u Õ 1 v3 sPy2 R uq , Ž 24c. u 1 v4 s P y 2 R u y , Ž 24d. u 1 1 v 5 s P q R u y q q 3Õ , Ž 24e. u Õ 1 1 v6 s P q R u y y y 3Õ , Ž 24f. u Õ where P, R and u,Õ are independent parameters. Finally, we substitute Eq. Ž23. and Eq. Ž24. into Eq. Ž22. and evaluate it using the Mathematica symbolic calculations package. As a result, we find that Eq. Ž22., and hence Eq. Ž19., do not hold on manifold Ž20.. wNote: We also verified that for T1234 ' 1, i.e. for the integrable NLS case, Eq. Ž22. does hold on manifold Ž20..x Consequently, I Ž6. must have a line singularity, which invalidates expansion Ž9.. Thus additional conserved quantities of the form Ž9. do not exist for Eq. Ž5.. The above consideration breaks down for d 0 s 0 Žcf. Eq. Ž12.., unless one has f Ž v . s A v q B, with

ž ž

/ /

ž ž

ž ž

/ /

/ /

71

A, B being arbitrary constants. Since the latter choice for f Ž v . with either A or B nonzero corresponds to the conservation of the mass andror the momentum, in which we are not interested here, we have to set f Ž v . s 0 to satisfy Eq. Ž12.. Then at the order O Ž a6 . we find the condition to be exactly of the form Ž19. where now I Ž4. has not yet been determined. The only nonzero solution to that equation which we have been able to find is the obvious one, i.e. Ž4. I1234 s T1234 . This corresponds to the Hamiltonian of Eq. Ž5. to be conserved. Furthermore, if we set Ž4. Ž6. I1234 s 0 and I123456 / 0, then at the order O Ž a8 . we find the condition

H Ž d d a.

1234,5678

. Ž IŽŽ 6567y34 . 34567 T128 Ž 12y8 .

Ž 6. Ž 123y78 . T Ž 56y4 . 456 . s 0 , yI12378

Ž 25 .

Ž6. IŽ567y34.34567 s I Ž6. Ž Ž v 5 q v6 q v 7 . y Ž v 3 q

where v4 ., v 3 , v4 , v 5 , v6 , v 7 ., etc. We verified that the most plausible choice for I Ž6. : Ž6. I123456 s

Ý

Ti jlŽ i jyl .TŽ m nyk . k m n ,

Ž 26 .

ijksP Ž123 . lmnsP Ž456 .

which satisfies symmetry conditions Ž11., does not satisfy Eq. Ž25.. Thus, we were unable to find any additional conserved quantities of Eq. Ž5. for d 0 s 0, either. 3. Periodically conserved quantities of Eq. (1) First, we rewrite Eq. Ž1. in the form similar to that of Eq. Ž5.: 1 ia˙v y d Ž z . v 2 av q d123 d 12,3 v a1 a 2 a 3) s 0 , 2 Ž 27 .

H

where d Ž z . s d 0 q D Ž z . and av s Ž1rŽ2 p .. ` Hy` dt expwyi vt x uŽt , z .. We look for a periodically conserved quantity in the form Ž9., with f Ž v ., I Ž4., I Ž6., etc. being independent of z. In Appendix B, we show that the same results as will be obtained below, also hold when these coefficients are periodic functions of z. ŽThe latter case includes, in particular, the Hamiltonian of Eq. Ž1... Thus, taking f Ž v ., I Ž4., I Ž6., etc. as being z-independent does not seem to limit the generality of our results. We show below

T.I. Lakobar Physics Letters A 260 (1999) 68–77

72

that Eq. Ž1. has no periodically conserved quantities of the form Ž9., apart from the mass and the momentum, which are, in fact, conserved exactly rather than periodically. In particular, we show, at the end of this section, that the Hamiltonian of Eq. Ž1. is not periodically conserved. Proceeding along the lines of the preceding section, we find at the order O Ž a 4 . a condition of the form Ž12. with T1234 ' 1 and d 0 being replaced by dŽ z .. Then the requirement 1 dI dz s 0 Ž 28 . 0 dz at that order yields the following equation for I Ž4. : z 1 1 Ž4. f 12 ,34 dz exp yi D12,34 d Ž zX . dzX 2 0 0

H

H

H

Ž4. Ž4. s i I1234 y 1. . Ž exp yid 0 D12,34

H

with cv being independent of z at this order. The Ž4. r.h.s. of Eq. Ž29. vanishes for d 0 D12,34 s 2p M, where M is any integer. Hence in order for a nonsingular I Ž4. to exist, the l.h.s. of Eq. Ž29. must vanish on any manifold of the form v 1 q v 2 y v 3 y v4 s 0, MgZ .

ž

Ž 31 .

For M s 0, this always holds, as discussed in Section 2, because f 12,34 s 0. For M / 0, f 12,34 / 0 in general. Moreover, it is easy to show that solutions to Eq. Ž31. always exist even for M / 0, such that, e.g., Ž v 1 y v 2 . 2 G maxŽ0,8p Mrd 0 .. Thus we must require that the integral on the l.h.s. of Eq. Ž29. vanish. This yields the following relation between the parameters of the dispersion map: 1 L1 q s N , Ž 32 . d0 where N is an arbitrary integer. In deriving Eq. Ž32. we have used Eq. Ž2. – Ž4X .. Thus, when condition Ž32. is satisfied, a nonsingular I Ž4. can always be found, and we need to proceed to the next order. Our consideration of the order O Ž a6 . is similar to that in Section 2, with one exception. Namely, in

/

Ž4. =exp yi D12 ,34

Ž 29 .

In deriving the last equation, we have used an approximation z i av s cv exp y v 2 d Ž zX . dzX , Ž 30 . 2 0

Ž4. d 0 D12,34 s 2p M ,

addition to the term which has the form of the l.h.s. of Eq. Ž18. with T1234 s 1 and d 0 replaced by dŽ z ., one also gets a contribution from the term which occured at the previous order Ži.e. the one of the form of the l.h.s. of Eq. Ž12.. That lower-order term contributes to the order O Ž a6 ., because cv in Eq. Ž30., which was used in the derivation of Eq. Ž29., does depend on z through terms of order O Ž a 2 . and higher Žcf. Eq. Ž27... Calculation of such terms would be a delicate task, due to the occurrence of a logarithmically diverging phase w16x, and we do not perform it here. Instead, we use the following simple trick. Let us denote f 12 ,34 Ž4. Ž4. FŽ z. s y d Ž z . D12 ,34 I1234 2 z

X

H0 d Ž z . dz

X

.

Ž 33 .

Then the z-integral over one map period of the l.h.s. of the counterpart of Eq. Ž12. for Eq. Ž1. can be rewritten as follows: 1

H0 dzH Ž d d c .

12 ,34 F

Ž z. E Ž c1 c 2 c3) c4) .

1

H0 dzH Ž d d .

sy

12 ,34

Ez

z

X

X

H0 F Ž z . dz . Ž 34 .

Here we have used integration by parts and the fact that H01 F Ž z . dz s 0 Žcf. Eq. Ž29... Also, the notation Ž d d c .12,34 is analoguous to the notation Ž d d a.12,34 , defined after Eq. Ž9.. Now the term E Ž c1 c 2 c 3) c 4) .rE z can be computed using Eqs. Ž30. and Ž27.. Adding the result to the term of the form of the l.h.s. of Eq. Ž18., as discussed above, we finally arrive at the condition at the order O Ž a6 .: 2 Re

1

H0 dzH Ž d d c .

123,456

z

ž

= f12,6 Ž 12y6 . z

Ž4. 12,6 Ž 12y6 .

H0 exp yi D

X

=

H0 d Ž z

XX

4. . dzXX dzX y2 i I ŽŽ45y3 . 345

. =exp yi DŽŽ 445y3 . 3,45

z

X

H0 d Ž z . dz

/

X

Ž6. s y Ž d d c . 123,456 I123456

H

Ž 6. = Ž exp yid 0 D123,456 y 1. .

Ž 35 .

T.I. Lakobar Physics Letters A 260 (1999) 68–77

Note that the r.h.s. of this equation is always real due to symmetry conditions Ž11.. Singularities of I Ž6. can occur on any of the following singular manifolds:

v 1 q v 2 q v 3 y v4 y v 5 y v6 s 0, MgZ .

Ž6. d 0 D123,456 s 2p M ,

Ž 36 .

On these manifolds, the l.h.s. of Eq. Ž35. is transformed into the form: Im

HŽ dd c .

ž

1

= y2

sin =

123,456 f 123 ,456

N 2

ž

/

y d 0 q D1

d0 q D2

Ž4. Ž 12y6 . d 0 D12,6

sin

qd M ,0

2

1

L1 d 0 q D1

q

1 2

sin

Ny1 2

H

Ž4. d 0 D12 ,6 Ž 12y6 .

L2 d0 q D2

/

small v and subsequently convince oneself that the result is indeed nonzero. We performed the verification using both ways. Thus we have shown that even with condition Ž32. satisfied, a nonsingular I Ž6. does not exist. Hence Eq. Ž1. does not have additional periodically conserved quantities for d 0 / 0. As in Section 2, the above analysis requires some modification for the case d 0 s 0. Since that modification is a straightforward combination of the calculations used in this and the previous sections, we do not give its details here, but simply state the final result. Namely, we were unable to find additional periodically conserved quantities in this case, either. To conclude this section, we now show that the Hamiltonian of Eq. Ž1. is not periodically conserved. The quadratic part of the Hamiltonian is: Hs dv

Ž4. Ž 12y6 . d 0 D12,6

1 Ž4. D12 ,6Ž12y6.

0

,

Ž 37 . where the integer N was defined in Eq. Ž32., and d M ,0 s 1 for M s 0 and d M ,0 s 0 otherwise. In deriving expression Ž37., we have also used Eq. Ž2. – Ž4X . and Ž29.. Now it sufficies to show that expression Ž37. does not vanish for M s 0 Ži.e. on singular manifold Ž20.., provided that f Ž v . / A v 2 q Bv q C. In fact, the term with d M ,0 does vanish there, as noted in the paragraph following Eq. Ž24.. However, the rest of the expression does not vanish on manifold Ž20.. The corresponding calculations, which require symmetrization of that term as per Eq. Ž22. and then use of Eq. Ž24., are too cumbersome even for Mathematica. The easiest way to verify that the term in question does not vanish is to simply evaluate it numerically Žstill with Mathematica or a similar package. for some particular values of d 0 and N and at some particular point on manifold Ž20.. Alternatively, one could expand it in the Taylor series for

73

ž

1 2

d Ž z . v 2 < av < 2 q PPP ,

/

Ž 38 .

i.e. in terms of expansion Ž9., f Ž v . s dŽ z . v 2r2. As shown in Appendix B, the same results are obtained for f Ž v . s v 2r2, and hence we present the details of our analysis for the latter form of f. We first consider the case d 0 / 0. If condition Ž32. is not satisfied, then the I Ž4. found from Eq. Ž29. is singular on any of the manifolds Ž31. with M / 0, and therefore the Hamiltonian is not conserved already at order O Ž a 4 .. If condition Ž32. is satisfied, then a nonsingular I Ž4. does exist, and at the next order, O Ž a6 ., we arrive at expression Ž37. where f 123,456 s Ž6. D123,456 . To guarantee existence of a nonsingular I Ž6., this expression must vanish on any of the manifolds Ž36.. For M s 0, it does indeed vanish due to the special form of f Ž v .. However, for M / 0, where f 123,456 / 0, it does not vanish identically. We verified this by taking random points on the following submanifold of manifold Ž36.: v 1 s v 2 s 0,

v3s

1 2p M v4 ,5 s " a, 2 d0

v6 s

2p M

2p M

q d0

d0

2p M

2p M

q d0

ž / ž /

2

d0

3 a2q , 4

2

a2y

1 4

Ž 39 . Žwhere a is a real parameter. and evaluating the symmetrized, as per Eq. Ž22., expression Ž37. for various values of N and M. Thus we showed that

T.I. Lakobar Physics Letters A 260 (1999) 68–77

74

the corresponding I Ž6. is singular even for f Ž v . s v 2r2, and hence there exists no periodically conserved quantity of Eq. Ž1. whose leading-order part would coincide with that of the Hamiltonian. In particular, the Hamiltonian itself is not conserved. In the case d 0 s 0 the analysis is modified as follows. First of all, it easy to see that f Ž v . must be identically zero. Then the first nontrivial condition arises at the order O Ž a6 .: 1

H0 dzH Ž d d c .

Ž4. Ž4. Ž 12y6 . exp yi D Ž 45y3 . 3,45 yI126

z

X

H0 d Ž z . dz 1

X

H0 d Ž z . dz

X

/

Ž4. exp yi D12 ,34 y 1 Ž4. Ž yi . D12 ,34

1

z

Ž4. 34 ,5 Ž 34y5 .

H0 exp yi D

=

ž

1234,5678

Ž 41 .

. I ŽŽ 434y5 . Ž567y34 .67

z

X

XX

. dzXX dzX

XX

. dzXX dzX

Ž 49 .

. yIŽŽ 4567y34 . 47 Ž56y3 .

z

. exp yi DŽŽ 456y3 . 3,56

=exp

s

ž

Ž4. yi D12 ,8 Ž 12y8 .

1

1 y

D1

ž

D2

/

Re

z

z

/

Re

HŽ dd c .

1234,5678

. Ž4. I ŽŽ 4567y34 . 47 Ž56y3 . Ž exp yi D Ž 56y3 . 3,56 y 1 . Ž4. Ž4. DŽ56y3.3 ,56 Ž DŽ56y3.3 ,56 q 2 p M .

Ž4. Ž4. Ž 34y5 . y 1 . IŽ34y5.Ž567y34.67 Ž exp yi D34,5 Ž4. Ž4. D34 ,5 Ž 34y5 . Ž D34,5 Ž 34y5 . q 2 p M .

/

.

Ž 43 . Obviously, on the 6-dimensional manifold

Ž4. D12 ,8Ž12y8. s 2 p M / 0 ,

Ž 44 .

one has enough degrees of freedom to make expression Ž43. nonzero, with the only exception being the case of a symmetric dispersion map when D 1 s yD 2 . Note that in the latter case, I Ž6. does not vanish identically, because expression Ž43. holds only on manifold Ž44.. Therefore, even if there exists a periodically conserved quantity with I Ž4. given by Eq. Ž41. and I Ž6. found from Eq. Ž42., it must be different from the Hamiltonian of Eq. Ž1. with d 0 s 0 Žnote that the latter also has f Ž v . ' u 0.. We did not investigate the possibility of existence of such a quantity in the case of a symmetric map, because the required calculations at the order O Ž a10 . are extremely cumbersome.

X

H0 d Ž z X

H0 d Ž z . dz

HŽ dd c .

. = I ŽŽ 6567y34 . 34567

D 22

v 1 q v 2 q v 3 q v4 y v 5 y v6 y v 7 y v 8 s 0,

.

H0 d Ž z

ž

y

s0 .

Note that at this order, we have obtained the same result as for Eq. Ž5. with d 0 s 0 Žcf. Eq. Ž23... However, at the next order we already find a difference from the case of Eq. Ž5.. At that order, O Ž a8 ., we obtain the following condition:

H0 dzH Ž d d c .

1

D 12

y

Its obvious nonzero solution is

H0

1

X

Ž 40 .

=

ž

=

ž

2 Re

2

123,456

. Ž4. = I ŽŽ 445y3 . 345 exp yi D12,6 Ž 12y6 .

Ž4. I1234 s

Ž4. where I Ž4. is given by Eq. Ž41.. For D12,8Ž12y8. s Ž6. 2p M, where M is a nonzero integer, I is singular, unless the l.h.s. vanishes there. Using the above Ž4. condition for D12,8Ž12y8. , the l.h.s. of Eq. Ž42. is transformed into the form:

/

4. Summary and discussion

X

1234,5678

Ž4. exp yi D12 ,8 Ž 12y8 . y 1 Ž4. Ž 12y8 . D12,8

/

,

Ž 42 .

The main results of this study are the following. First, we showed Žin Appendix A. that the Painleve´ test fails for Eq. Ž1., thus indicating that that equation is not integrable by the Inverse Scattering Transform. Second, we applied the method of Zakharov and Schulman to the reduced form of Eq. Ž1. in the strong DM limit, Eq. Ž5., and showed that it does not have conserved quantities other than the mass Žnum-

T.I. Lakobar Physics Letters A 260 (1999) 68–77

ber of photons., the momentum, and the Hamiltonian. Third, using the same method, we showed that Eq. Ž1. does not have periodically conserved quantities other than the mass and the momentum, which are conserved exactly rather than periodically. We note here that there was a slight difference between the calculations for Eq. Ž5. and those for Eq. Ž1.. While the former calculations are just the straightforward application of the Zakharov–Schulman method to the specific equation, the latter ones required a certain modification, which was explained in the paragraph immediately preceding Eq. Ž33.. Let us also mention that our results for the case d 0 s 0, indicating the absence of conserved Žor periodically conserved. quantities for Eqs. Ž1. and Ž5., are not truly rigorous. Rather, we showed that for some very plausible form of I Ž4. and I Ž6., additional conserved Žor periodically conserved. quantities do not exist. Establishing a rigorous result in that case remains an open problem. We also showed that when condition Ž32. holds, then Eq. Ž1. has infinitely many quantities with quadratic leading-order part, which are periodically conserved up to the order O Ž a 4 . inclusively. This fact can be interpreted as ‘approximate integrability’ in those cases where one is justified to treat the nonlinear terms in Eq. Ž1. as a small perturbation. This is precisely what occurs in the theory of the non-return-to-zero ŽNRZ. data transmission in optical telecommunications. There, a logical ONE is represented by a rectangular pulse which occupies the entire bit slot. If there are two adjacent ONEs, the field between them does not go to zero, in contrast to what occurs in the soliton transmission format. Hence the name ‘NRZ’. An important quantity in optical telecommunications is the detected pulse energy Žwhich we called ‘mass’ or ‘number of photons’ above.. Thus, since an NRZ pulse is approximately 5 times broader than the soliton which would represent the same bit of information, then its power Ži.e. < u < 2 . is correspondingly lower. Therefore, one usually treats nonlinear terms in the evolution of NRZ pulses as a small perturbation. One of the serious detrimental effects caused by collisions of NRZ pulses belonging to different frequency channels with frequencies v 1 and v 2 is four-wave mixing ŽFWM., i.e. creation of a relatively small field at frequency, say, 2 v 1 y v 2 ,

75

through the nonlinearity. If in addition to the frequency matching condition,

v1 q v1 s v2 q Ž2 v1 y v2 . ,

Ž 45a.

the propagation constants, bv s dŽ z . v 2r2, of these four waves satisfy the relation 1 2

d 0 Ž v 12 q v 12 y v 22 y Ž 2 v 1 y v 2 . s 2p M ,

MgZ ,

2

. Lmap Ž 45b.

then the FWM field with frequency Ž2 v 1 y v 2 . will grow linearly Žon average. with the propagation distance w17x. Note that Eq. Ž45. coincide with Eq. Ž31. above. Experimentally, one observes w17,18x sharp peaks of the FWM field, whose locations are determined by Eq. Ž45b., as one varies the frequency separation Ž v 1 y v 2 . Žnote that the l.h.s. of Eq. Ž45b. equals yd 0 Ž v 1 y v 2 . 2 .. However, if parameters of the dispersion map satisfy condition Ž32., then we predict that Žalmost. no FWM will be observed for any frequency separation. This conclusion, which in our analysis is a consequence of the aforementioned ‘approximate integrability’ of Eq. Ž1., is confirmed by the direct calculation of the FWM field w19x. In fact, our condition Ž32. follows from Eq. Ž8. of Ref. w19x, where one has to set N s 2 and a s 0. Note also that the same conclusion regarding the absence Žor, more precisely, strong suppression. of FWM in collisions of NRZ pulses should also hold in the strong DM regime irrespective of the exact values of d 0 and L1 , because the corresponding equation, Eq. Ž5., is also ‘approximately integrable’ up to the order O Ž a 4 .. Since condition Ž32. implies ‘integrability’ of Eq. Ž1. only at the lowest nontrivial order Ži.e., at O Ž a4 .., we expect that it should not be relevant for the dynamics of DM solitons. Indeed, the latter, in contrast to NRZ pulses, are essentially nonlinear objects, for which all terms O Ž a n . in expansion Ž9. play equally important role. Thus, DM solitons should exhibit features of solitary waves in non-integrable systems. One such feature is the inelastic interaction of adjacent DM solitons Žwith the same central frequencies., which was reported in Ref. w20x. Our own numerical simulations of Eq. Ž1., performed with condition Ž32. being imposed on the coefficients DŽ z . and d 0 , showed that the interaction remains inelastic under that condition as well.

T.I. Lakobar Physics Letters A 260 (1999) 68–77

76

More interesting, however, is the problem of FWM in collisions of DM solitons with disparate frequencies. On one hand, the results of Ref. w21x, where this problem was considered for the conventional Ži.e. NLS. solitons, suggest that a treatment based on the same assumptions as in the NRZ case Žcf. their Eq. Ž4.., except that the shape of the soliton is now explicitly accounted for, yields good agreement with numerical simulations. On the other hand, since the soliton width is considerably smaller than the distance between consecutive solitons in a communication channel, there is, in general, asymmetry between the beginning and the end of the collision w22x in systems with periodically varying dispersion. This asymmetry, and hence the size of the FWM field, depends on a number of parameters, such as the pulse width, frequency separation, average dispersion, and lengths L1 and L2 Žs Ž1 y L1 .. of the fiber sections. We numerically simulated the collision of two DM solitons of Eq. Ž1., having fixed all of these parameters Žat, respectively, T0 s 1r2 Žcf. Ref. w5x., v 1 y v 2 s 5.8p , and d 0 s 1r17.5. but L1 , which we varied between 0 and 1. The dependence of the energy of the FWM field appeared to be both irregular and strong, with variations between the maximum and minum values being more than two orders of magnitude. This numerical observation, along with the above argument involving Ref. w21x, indicate that calculation of the FWM field in collisions of DM solitons is an interesting open problem.

'

where fstyc Ž z. ,

The author gratefully acknowledges many useful discussions with V. Zharnitsky. This research was supported by the National Science Foundation under grant PHY94-15583 and by the Office of Naval Research under grant N00014-98-1-0630.

1

u2 s

3Ž d Ž z . .

Õ2 s

`

usf a

Ý ujf j, js0

`

Õsf b

Ý Õj f j , js0

Ž A1.

Ž A2.

3Ž d Ž z . .

2

ž ž

i

d Ž u0 d Ž z . . dz

yi

y u0

d Ž Õ0 d Ž z . . dz

2

dc

ž // ž // ,

dz

dc

y Õ0

2

dz

,

Ž A6. 1 Õ3 s

u 20

ž

u3 d Ž z . q

u0

2

dc

d Ž z . dz 2

/

.

Ž A7. Finally, at the order O Ž f . we find that the necessary condition for u 4 and Õ4 to exist is 1

q

Following the approach outlined, e.g., in Ref. w23x, we consider the following independent expansions for u and Õ ' u ) :

2

1

Ž dŽ z . . Appendix A. Painleve´ test for Eq. (1)

Õj s Õj Ž z . .

Coefficients a , b and functions u j , Õj , and c Ž z . are to be determined. Since the details of this approach can be found, e.g., in Ref. w23x, here we only give final results obtained at each step of the calculations. Substituting Eq. ŽA1. into Eq. Ž1., we obtain the following condition at the leading order: a s b s y1, u 0 Õ 0 s yd Ž z . . Ž A3. jy3 . Ž Next, at the order O f we find that the resonances w23x are located at j s y1, 0, 3, 4 . Ž A4. Note that the first two resonances correspond to the arbitrariness of c Ž z . and u 0 , respectively. Therefore, we have to consider all the conditions arising at each successive order from O Ž f 1y 3 . s O Ž fy2 . to O Ž f 4y 3 . s O Ž f .. At the orders O Ž fy2 . through O Ž f 0 . we find, respectively, the following conditions: iÕ 0 d c iu 0 d c u1 s , Õ1 s y , Ž A5. d Ž z . dz d Ž z . dz

u 3 arbitrary,

Acknowledgements

uj suj Ž z . ,

2

ž

3u 0 d Ž z .

du 0 d Ž d Ž z . . dz

dz

d2 dŽ z . dz 2

y 3u 0

ž

d dŽ z . dz

2

/

2

/

s0 .

Ž A8.

Eqs. ŽA7. and ŽA8. were obtained with the Maple symbolic calculations package. Obviously, Eq. ŽA8. does not hold for the piecewise-constant dispersion coefficient dŽ z .. Thus Eq. Ž1. does not pass the Painleve´ test.

T.I. Lakobar Physics Letters A 260 (1999) 68–77

Appendix B. Coefficients in expansion (9) for periodically conserved quantities Here we show that all conclusions of Section 3 remain the same if instead of constant coefficients f Ž v ., I Ž4., I Ž6., etc., considered there, one takes those coefficients to be periodic functions of z. Naturally, we assume that the periodicity is that of the dispersion map. Consider the quantity H01 Ž dIrdz . dz. Using Eq. Ž9., where now the coefficients are functions of z, we find: 1

H0

dI dz

dz s d v Ž f Ž v , zs1 . < av Ž zs1 . < 2

H

yf Ž v , zs0 . < av Ž zs0 . < 2 . q PPP s

1

H0 dzHd v f Ž v , z s 0.

d dz

< av < 2 q PPP .

Ž B1. We have not written explicitly the terms proportional to I Ž4., I Ž6., etc., because they have exactly the same form as the term in Eq. ŽB1.. We have also used the periodicity condition: f Ž v , z s 1. s f Ž v , z s 0.. Thus the statement formulated at the beginning of this Appendix is proved.

References w1x N.J. Smith, N.J. Doran, W. Forysiak, F.M. Knox, J. Lightwave Technol. 15 Ž1997. 1808. w2x S.K. Turitsyn, V.K. Mezentsev, E.G. Shapiro, Opt. Fiber Technol. 4 Ž1998. 384.

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w3x T.I. Lakoba, J. Yang, D.J. Kaup, B.A. Malomed, Opt. Commun. 149 Ž1998. 366. w4x J.H.B. Nijhof, N.J. Doran, W. Forysiak, F.M. Knox, Electron. Lett. 33 Ž1997. 1726. w5x T.I. Lakoba, D.J. Kaup, Electron. Lett. 34 Ž1998. 1124; T.I. Lakoba, D.J. Kaup, Phys. Rev. E 58 Ž1998. 6728. w6x S.K. Turitsyn, V.K. Mezentsev, JETP Lett. 67 Ž1998. 640 wPis’ma Zh. Eksp. Teor. Fiz. 67 Ž1998. 616x; S.K. Turitsyn, T. Schafer, V.K. Mezentsev, Opt Lett. 23 Ž1998. 1351. ¨ w7x C. Pare, ´ Opt Commun. 154 Ž1998. 9. w8x M.J. Ablowitz, G. Biondini, Opt. Lett. 23 Ž1998. 1668. w9x I.R. Gabitov, S.K. Turitsyn, Opt. Lett. 21 Ž1996. 327. w10x S.B. Medvedev, S.K. Turitsyn, JETP Lett. 69 Ž1999. 499. w11x V.E. Zakharov, E.A. Kuznetsov, Physica D 18 Ž1986. 455. w12x V.E. Zakharov, E.I. Schulman, Physica D 1 Ž1980. 192. w13x V.E. Zakharov, E.I. Schulman, Physica D 29 Ž1988. 283. w14x V.E. Zakharov, E.I. Schulman, in: V.E. Zakharov ŽEd.., What is Integrability?, pp. 185–250, Springer-Verlag, New York, 1991; V.E. Zakharov, A. Balk, E.I. Schulman, in: A.S. Fokas, V.E. Zakharov ŽEds.., Important Developments in Soliton Theory, pp. 375–404, Springer-Verlag, New York, 1993. w15x V.E. Zakharov, E.I. Schulman, Physica D 4 Ž1982. 270. w16x A.I. Dyachenko, Y.V. Lvov, V.E. Zakharov, Physica D 87 Ž1995. 233. w17x K. Inoue, Opt. Lett. 17 Ž1992. 801. This work considers FWM in a system with periodically compensated loss rather than with periodic dispersion. However, its main conclusion is also valid for the periodic dispersion case. w18x E.A. Golovchenko, N.S. Bergano, C.R. Davidson, Photon. Technol. Lett. 10 Ž1998. 1481. w19x K. Inoue, H. Toba, J. Lightwave Technol. 13 Ž1995. 88. w20x T. Yu, E.A. Golovchenko, A.N. Pilipetskii, C.R. Menyuk, Opt. Lett. 22 Ž1997. 793. w21x M.J. Ablowitz, G. Biondini, S. Chakravarty, R.B. Jenkins, J.R. Sauer, Opt. Lett. 21 Ž1996. 1646. w22x A.M. Niculae, W. Forysiak, A.J. Gloag, J.H.B. Nijhof, N.J. Doran, Opt Lett. 23 Ž1998. 1354; H. Sugahara, A. Maruta, Y. Kodama, Opt Lett. 24 Ž1999. 145. w23x M.J. Ablowitz, P.A. Clarkson, Solitons, Nonlinear Evolution Equations and Inverse Scattering, Chap. 7, Cambridge Univ. Press, Cambridge, 1991.