LIGHT SOLITONS IN NONLINEAR MEDIA. LIGHT PROPAGATION IN

Bulg. J. Phys. 29 (2002) 51–69

LIGHT SOLITONS IN NONLINEAR MEDIA.LIGHT PROPAGATION IN FIBER-LIKE MEDIA

KH. I. PUSHKAROVDepartment of PhysicsHigher Institute for Civil Engineering and Architecture1 Smirnenski Str., 1421 Sofia, Bulgaria

Abstract. The light beam behaviour in nonlinear media is consideredin the case when the nonlinear polarization contains susceptibilities ofthird and fifth order. A number of soliton solutions of the nonlineardifferential equations describing self-action effects is obtained in ana-lytical form. The existence of the solitary pulses obtained is discussedfrom energetically point of view.

PACS number: 42.65.Tg, 42.65.Sf, 05.45.Yv

1. Introduction

In the paper [1] some nonlinear light phenomena with regard to the high powerlight density have been considered. For dielectric media which possess a centreof symmetry the polarization P has been presented in the form

P = κE + χ(3)E3 + χ(5)E5 (1)

where E denotes the electric field, κ is the linear susceptibility and χ(3) andχ(5) are the nonlinear ones of third and fifth order, respectively. Using (1)together with Maxwell’s equations, the following nonlinear equation of motionwas obtained in [1](

∇2 − 1c2

∂2

∂t2

)E =

4π

c2

[κ

∂2E∂t2

+ χ(3) ∂2E3

∂t2+ χ(5) ∂2E5

∂t2

]. (2)

If the radiation field is taken to be of the form

E = E(r, t) ei(kz−ωt) + c.c. , (3)

1310–0157 c© 2002 Heron Press Ltd. 51

Kh. I. Pushkarov

52

Light Solitons in Nonlinear Media. Light Propagation in Fiber-Like Media

where

U = − c2

4πω2

[(ω2

c∗2− k2)|E |2 +

2πσω2

c2(3χ(3)|E |4 +

203

ρχ(5)|E |6)](10)

plays the role of a potential energy. The light propagation will be considered asa motion determined by the extremum of U with respect to |E|. Such a mannerthe condition

dU

d|E |2 = 0 (11)

together with the series expansion (1) (in which the E5-term is supposed to besmaller than E3-one), shall lead to some restrictions upon the parameters ofthe solutions obtained. In particular, (11) leads to

|E |2 =3

20ρ

∣∣∣∣∣χ(3)

χ(5)

∣∣∣∣∣ (1 ±√1 − Q) (12)

where

Q ≡ 10ρχ(5)(ω2 − k2c∗2)c2

9π(χ(3))2ω2c∗2. (13)

(At that sign plus in (12) has to be used for which U = Umin). On the otherhand the nonlinear terms in (8) lead to

|Emax| <

√3

10ρ

∣∣∣∣χ(3)

χ(5)

∣∣∣∣ (14)

which must be satisfied if E5-term is smaller compared to E3-one [1].As in the previous considerations [1] the most interesting cases χ(3) > 0,

χ(5) < 0 and χ(3) < 0, χ(5) > 0 will be considered separately.

1.1. Case χ(3) > 0, χ(5) < 0

In this case Eq. (8) has the following soliton solutions.

1.1.1. Kink (anti-kink) solution

E(z, t) = E0 ei(qz−Ωt+ϕ) sinhz − z0 − vt

L×(1 + sec2 η sinh2 z − z0 − vt

L)− 1

2(15)

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Kh. I. Pushkarov

where

q = −k − Δq ,

Δq = ± kβ∗√

b∗2 − 1

√1 − ω2

k2c∗2(β∗ ≡ v

c∗> 1) , (16)

Ω = −ω − ΔΩ, ΔΩ = ±√

k2c∗2 − ω2

β∗2 − 1, (17)

E0 = ± 32

tan

√χ(3)

5ρ|χ(5)|(2 − cos 2η), (18)

L =2c(2 − cos 2η)3ωχ(3) sin 2η

√10ρ|χ(5)|(2 − cos2η)

3πσ, (19)

η = arccos(

3√

1 − Q

2√

1 − Q + 1

) 12

. (20)

The soliton (15) represents a kink (sign plus in (14)) or anti-kink (sign minusin (14)) which moves along the z-axis with a velocity v greater than the lightphase velocity c∗ in the medium without dispersion or radiation (β∗ ≡ v/c∗ >1). Naturally, it must be v < c (c being the light velocity in vacuum), i. e.

c∗ < v < c∗√

1 + 4πκ . (21)

The wave vector component q as well as the frequency Ω also depend on v.At that the kink frequency ΔΩ can be very small. The parameter L character-izes the length of the formation (15). In fact 2L plays the role of the effectivewidth of the kink. The constant z0 appears as a result of the translational in-variance of the system and represents the center of the solitary formation (15);ϕ is a phase constant.The parameter η is determined with the help of the extremum condition (11).

Taking into account that U has a minimum when |E |2 takes its maximal value

|E |2 = E0

2cos2 η

and using (12) (with sign plus, which corresponds to Umin) the expression (20)can be obtained. The condition (14) leads to the inequality

0 < k2c∗2 − ω2 <9πσ(χ(3))2ω2c∗2

10ρ|χ(5)|c2(22)

54


which must be satisfied. It must be noted that Eq. (8) has also a solution of thetype

E1(z, t) = E0 ei(qz−ωt+ϕ) (23)

where q, Ω, E0 and η have the same values as the kink (15). To compare(15) and (23) we shall calculate the energies W (for the kink) and W1 (for theplane-wave (23)). Using the energy density (14), we find that

W =+∞∫

−∞W (z, t) dz (24)

contains divergent terms. The calculation of W1 leads to an expression whichcontains the same divergent terms as W . Thus the energy difference

ΔW ≡ W − W1 = − 3c3η

2πωc∗2√

30π|χ(5)|(β∗2 − 1)

×[k2c∗2

ω2− 1 ∓

√(k2c∗2

ω2− 1)

(β∗2 − 1)

+27πc∗2σ(χ(3))2

8c2ρ|χ(3)|(β∗2 − 1)1 − 2 cos 2η

(2 − cos 2η)2

(1 +

sin 2η

2η

2 − cos 2η

1 − 2 cos 2η

)].

(25)

The analysis of (25) shows that ΔW can be negative. Thus the solitary for-mation (15) can be energetically more favourable than the plane-wave (23).For the case of Rb-vapor with atom density N = 108 cm−3, assuming the

atomic susceptibilities to be χ(3) = 4×10−31 esu and χ(5)at = −2.1×10−41 esu

[2] and taking the light wavelength λ = 1.06×10−4 esu and η ≈ 0.52, weobtain

E0 = 4.3×10−4 esu , L = 8.5√

ρ

σ(β∗2 − 1) cm.

1.1.2. “Bell” soliton solution

E(z, t) = E0 ei(qz−Ωt+ϕ) coshz − z0 − vt

L×(cosec2 η cosh2 z − z0 − vt

L − 1)− 1

2(26)

where

q = −k − Δq ,

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Kh. I. Pushkarov

Δq = ± kβ∗√

b∗2 − 1

√1 − ω2

k2c∗2− 4(β∗2 − 1)

k2L2+

R

k2, (27)

Ω = −ω − ΔΩ , ΔΩ = vΔq , (28)

R ≡ ω2

c∗2− k2 +

12πχ(3)ω2 sin2 η

c2E2

0 − 40πρ|χ(5)|ω2 sin4 η

c2E4

0 , (29)

E0 = ± cot η

2

√3χ(3)

5ρ|χ(5)| , (30)

L =c

ωχ(3) sin 2η

√10ρ|χ(5)|(β∗2 − 1)

3πσ, (31)

β∗2 ≡ (v

c∗)2 > 1 . (32)

As one can see from (26) the bell soliton will exist if

cosec2 η > 1 . (33)

The analysis of the extremal condition (11) shows that taking into account(33) the electric amplitude must be of the form E(r, t) eiϕ, where the phaseconstant ϕ can be taken the same as in (26). Such a manner the value

E(z, t) =[

3χ(3)

20ρ|χ(5)|(

1 −√

1 − 10ρ|χ(5)|(k2c∗2 − ω2)c2

9πσ(χ(3))2ω2c∗2

)] 12

(34)

corresponds to the extremum (minimum) of U . On the other hand, as it can beseen from (26), |E(z, t)| takes the value E0 sin η at infinity (where (26) turnsinto a plane wave). Hence, from (30) and (34) the following expression for ηcan be obtained:

η = arcsin[1 − 10ρ|χ(5)|(k2c∗2 − ω2)c2

9πσ(χ(3))2ω2c∗2]

14 . (35)

Also, the condition (14) leads to the inequality

0 < k2c∗2 − w2 <9πσ(χ(3))2ω2c∗2

10ρ|χ(5)|c2. (36)

One can see that for η given by (35) R = 0. According to the considerationmade in Sec. 3 of [1] the condition R = 0 plays the role of the dispersionrelation between ω, k and E0. The substitution of R = 0 into (27) and (28)

56


gives the final forms of Δq and ΔΩ:

Δq = ± kβ∗√

β∗2 − 1

√1 − ω2

k2cc∗2 − 4(β∗2 − 1)k2L2

, (37)

ΔΩ = vΔq . (38)

The condition R = 0 coupled with (36) gives the following final form of theinequality which must be satisfied:

3πσ(χ(3))2ω2c∗2

10ρ|χ(5)|c2sin2 2η < k2c∗2 − ω2 <

9πσ(χ(3))2ω2c∗2

10ρ|χ(5)|c2. (39)

As it is easy to see, the condition (14) is always satisfied.Equation (8) has also a homogeneous plane-wave solution of the form

E(z, t) = E0 ei(qz−Ωt+ϕ) (40)

with the same parameters as the bell soliton (26). The subtraction of the energyW1, corresponding to (40), from the energy W (corresponding to (26)) leadsto the result:

ΔW ≡ W − W1 = − 3k2c3η

πω3√

30πσρ|χ(5)|(β∗2 − 1)

×[1 − ω2

k2c∗2(1 +

3πσ(χ(3))2 sin2 2η

10ρ|χ(5)| )

± ω

kc∗

√√√√[1 − ω2

k2c∗2

(1 +

3πσ(χ(3))2 sin2 2η

10ρ|χ(5)|

)](β∗2 − 1)

+3πσ(χ(3))2ω2(2 − β∗2)

80ρ|χ(5)|k2c2

(1 +

sin 2η

2η

2 − 5 cos 2η

1 + 2 cos 2η

)].

(41)

The analysis of (41) shows that ΔW can be negative. Hence, the solitonformation (26) can be more favourable than (40). Besides, the soliton canpropagates with a velocity v higher than the phase velocity c∗ in the mediumconsidered (see (32)) and without “smearing”, i. e. in such nonlinear media thepropagation of wave packets with velocity v in the interval

c∗ < v < c

and without radiation takes place. Looking at (38) one can see that self-modulation effect takes place: the phase depends on the amplitude throughthe soliton length L and vice versa. For Rb-vapor, considered in the previous

57

Kh. I. Pushkarov

Subcase 1, when η ≈ π2the solitary formation (26) has a very small amplitude

and L ≈ 73.76√

ρ

σ(β∗2 − 1) cm.


E(z, t) = E0 ei(qz−Ωt+ϕ)

(1 + sech

z − z0 − vt

L) 1

2

(42)

(compare with Eq. (58) in [1], where

q = −k ± Δq ,

Δq =β∗

c∗

√k2c∗2 − ω2

β∗2 − 1, (β∗ > 1) , (43)

Ω = −ω ± ΔΩ , ΔΩ =c∗

β∗ Δq , (44)

E0 = ± 34

√χ(3)

5ρ|χ(5)| , (45)

L =c∗

2

√β∗2 − 1

k2c∗2 − ω2, (46)

k2c∗2 − ω2 =27πσ(χ(3))2ω2c∗2

32ρ|χ(5)|c2. (47)

Equation (8) has also the plane-wave solution

E1 = E0 ei(qz−Ωt+ϕ) (48)

with the same parameters as for (42).In this case the energy difference ΔW ≡ W − W1 is equal to

ΔW =3c3

8ω3√

30πσρ|χ(5)|c2

×⎡⎣ 9π + 4

10π(k2 − ω2

c∗2) ∓√

(k2 − ω2

c∗2)(β∗2 − 1)

⎤⎦ .

(49)

It is easy to see that ΔW > 0, i. e. in this case the plane-wave (48) is ener-getically more favourable than the soliton.

58


1.1.4. “Inverse bell” soliton solution


(1 − sech

z − z0 − vt

L) 1

2

(50)

where q, σ, E0 and L are given by (43), (44), (45) and (46), respectively.Taking into account that (48) is also solution of (8), for the energy differenceΔW = W − W1 one obtains:

ΔW = − 3c3

8ω3√

30πσρ|χ(5)|(β∗2 − 1)

×[9π − 410π

(k2 − ω2

c∗2) ± ω

c∗

√(k2 − ω2

c∗2)(β∗2 − 1)

].

(51)

Obviously ΔW < 0, i. e. the solitary formation (50) is energetically morefavourable than the plane-wave. The soliton (50) moves with velocity v > c∗

(c∗ < v < c).When the field amplitude varies slowly in time so as∣∣∣∣∣ ∂

2E∂t2

∣∣∣∣∣� ω

∣∣∣∣∣ ∂E∂t

∣∣∣∣∣ (52)

is satisfied, (8) turns into the following nonlinear Schrödinger equation withnonlinearities of third- and fifth-power with regard to the function:

i

(2ω

c∗2∂E∂t

+ 2k∂E∂z

)+

∂2E∂z2

+

(ω2

c∗2− k2

)E

+4πσω2

c2(3χ(3)|E |2 + 10ρχ(5)|E |4)E = 0

(53)

which has the following soliton solutions when χ(3) > 0 and χ(5) < 0.

1.1.5. First type kink (anti-kink) solution


L×(1 + sech2 η sinh2 z − z0 − vt

L)− 1

2(54)

where

q = −k + Δq ,

Δq =ω

c∗β∗ , (55)

59

Kh. I. Pushkarov

Ω = −ω + ΔΩ , ΔΩ =ω

2(2 + β∗ − k2c∗2

ω2) , (56)

E0 = ± coth η

2

√3χ(3)

5ρ|χ(5)| , (57)

L =2c(cos 2η − 2)3ωχ(3) sinh 2η

√10ρ|χ(5)|

3πσ, (58)

η = arccosh

( 3√

1 − Q

2√

1 − Q − 1

) 12

, (59)

Q being given by (13). The condition (14) leads to the inequalities

0 < k2c∗2 − ω2 <27π(χ(3))2ω2c∗2

40ρ|χ(5)|c2. (60)

It must be noted that (54) differs from (15) by the replacement sec η → sech η.We do not repeat the procedure of finding η described also in Sec. 1 of [1].Taking into account that (53) has also a plane-wave solution of the type

E1(z, t) = E0 cosh η ei(qz−Ωt+ϕ) (61)

with the same parameters q, ω, E0, η as (55), (56), (57), (59), respectively, forthe energy difference ΔW ≡ W − W1 we obtain

ΔW = − 3c3η

4πc∗2ω√

30π|χ(5)|

[(β∗ +

2kc∗

ω)β∗ − 2k2c∗2

ω2

− 27πc∗2σ(χ(3))2(2 cosh 2η − 1)40c2ρ|χ(5)|(cosh 2η − 2)2

×(1 +

sinh 2η2η

cosh 2η − 22 cosh 2η − 1

)].

(62)

It must be noted that neglecting the second time derivatives of E , the param-eters of the soliton are changed drastically: trigonometric functions are changedby hyperbolic ones! Also ΔW > 0 when β∗ ≤ 1 and a soliton formation isnot energetically favourable in comparison with the plane-wave (61).For Rb-vapor (see above) Δq ≈ 2.2×105β∗ cm, ΔΩ ≈ 8.9(1 + β∗2) s−1,

E0 ≈ 6.6×104 esu, L ≈ 8.13 cm (when ω ≈ kc∗, η ≈ 1.146).

1.1.6. Second type kink (anti-kink) solution


(1 + e±

z−z0−vt

L

)− 12

(63)

60


where q is given by (55),

Ω = −ω + ΔΩ, ΔΩ =ω

2

(1 + β∗2 − 27πc∗2(χ(3))2

40ρc2|χ(5)|)

, (64)

E0 = ± 32

√χ(3)

10ρ|χ(5)| , (65)

L =c

3ωχ(3)

√10ρ|χ(5)|

3πσ(66)

and the inequality (14) is always satisfied.The signs plus and minus in (63) correspond to anti-kink and kink, respec-

tively. When v < c∗ the frequency of the kink ΔΩ ≈ ω/2 i. e. the generationof subharmonics is possible. Equation (53) has also a plane-wave solution ofthe form

E(z, t) = E0 ei(qz−Ωt+ϕ) (67)

with the same parameters q, Ω and E0 as (55), (64) and (65), respectively.The light can have the form of a step, e. g. “anti-kink-like step”, i. e. when−∞ < z − z0 − vt < 0 E1 is given by (67) and when 0 < z − z0 − vt < ∞,E1 = 0. The comparison of the energy W1 of such a formation with the energyW of the anti-kink (63) gives for the difference ΔW ≡ W − W1 the result

ΔW =81πσ(χ(3))2

640√

30πσω(ρ|χ(5)|) 32

> 0 .

Thus the step function is energetically more favourable than (63).It must be noted that a solution of the type (63) does not exist for the

“full equation” (8). Neglecting the second time derivative of the electric fieldwe can obtain solutions (in particular solitons) which cannot exist in reality.Nevertheless, we shall give such solutions which exist according to (52).

1.1.7. “Inverse bell” soliton solution of the form


L×(1 + cosech2 η cosh2 z − z0 − vt

L)− 1

2(68)

where q and Ω are given by (55) and (56), respectively,

E0 =32

coth η

√χ(3)

5ρ|χ(5)|(2 + cosh 2η), (69)

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Kh. I. Pushkarov

L =2c(2 + cosh 2η)3ωχ(3) sinh 2η

√10ρ|χ(5)|

3πσ, (70)

η = arcsin h

( 3√

1 − Q

1 − 2√

1 − Q

) 12

, (71)

Q being given by (13), and

27π(χ(3))2ω2c∗2

10ρ|χ(5)|c2< ω2 − k2c∗2 <

18π(χ(3))2ω2c∗2

5ρ|χ(5)|c2

has to be satisfied. Besides (68), Eq. (53) has a plane-wave soliton

E1(z, t) = E0 sinh η ei(qz−Ωt+ϕ) (72)

with q, Ω, E0 and η given by (55), (56), (59) and (71), respectively (the pa-rameters of the soliton formation above). The comparison of the energies W(related to (68)) and W1(related to (72)) gives

ΔW ≡ W − W1 = − 3c3η

4πωc∗2√

30πσρ|χ(5) |[(β∗2 − k2c∗2

ω2+

27πσc∗2(χ(3))2(1 + 2 cosh 2η)40c2ρ|χ(5)|(2 + cosh 2η)2

×(1 − sinh 2η

2η

2 + cosh 2η

1 + 2 cosh 2η

)].

(73)

The energy difference (73) can be negative (β∗ ≥ 1), i. e. (68) can be ener-getically more favourable than (72).In fact, the inverse bell soliton above is a “dark” soliton. For it

|E|2 ∼ cosh2 z−z0−vtL

1 + cosech2 η cosh2 z−z0−vtL

,

i. e. the light is expelled from the soliton region.



(1 + sech η cosh

z − z0 − vt

L)− 1

2

(74)

where q and E0 are given by (55) and (65), respectively,

Ω = −ω + ΔΩ, ΔΩ =ω

2

(1 + β∗2 − c∗2

ω2L2

), (75)

62


and

L =c coth η

3ωχ(3)

√10ρ|χ(5)|

3πσ. (76)

All of the quantities obtained have the same meaning as above but the para-meter η cannot be determined by minimization of the potential energy U (10).The parameter η can be connected with a given amplitude E0 (or the peak powerdensity of the light beam). Another possibility is to normalize the solution (74)according to the condition

N =+∞∫

−∞|E|2 dx dy dz

where N can be interpreted as a number of photons in the soliton region. Theintegration leads to

E0 =(

N

4σLcoth η

η

) 12

(77)

which gives

η = Nω

c

√10πρ|χ(5)|

3σ. (78)

On the whole E0 can be treated as a known quantity. In the case considered,the energy of the system W has the finite value

ΔW = − 3c3η

8πc∗2ω√

30πρ|χ(5)|

[1 − β∗2 − 27πc∗2σ(χ(3))2 tanh2 η

40c2ρ|χ(5)|

× [1 − 4 coth2 η +6 coth η

η2(1 − η coth η)]

].

(79)

The bell soliton (74) is favourable energetically if β∗ � 1.It must be noted that in this subcase the self-modulation effect also takes

place. The frequency Ω (75) depends on the soliton length L and

E20L2 =

c2

12πσω2χ(3). (80)

So the amplitude drives the phase and vice versa.Equation (75) shows also that if β∗ is small (v/c∗) ΔΩ can be changed

drastically (Ω → ω/2) and the effect of self-induced transparency can takeplace.

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Kh. I. Pushkarov

It must be noted also that the full equation (8) (with the second time deriva-tive) has not a solution which can be turned into (74) when the condition (52)and (53) are used (see the end of the Subcase 6 above). The condition (52)and Eq. (53) are self-consistent when β∗ � 1. The analysis made leads to thewell-known fact that neglecting of the higher derivative can drastically changethe physical results. As it was mentioned in [1] the presence of the second timederivative imposed also a restriction on the soliton propagation velocity. In thesolutions of the Schrödinger-like equations the velocity v is a free parameter.The analysis made here also has for an object to show that the use of such typeequations (such considerations occur in the literature) is illegal although veryoften the form of the solutions is apparently the same as for the case when thesecond time derivative is not neglected.

1.2. The Case χ(3) < 0, χ(5) > 0

In this case Eq. (8) has the following soliton solutions

1.2.1. Kink (anti-kink) solution


L×(1 + sec2 η sinh2 z − z0 − vt

L)− 1

2(81)

where

q = −k − Δq ,

Δq = ± kβ∗√

1 − β2

√ω2

k2c∗2− 1 , (82)

Ω = −ω − ΔΩ, ΔΩ = Δqc∗/β∗ , (83)

E0 = ± 32

tan η

√|χ(3)|

5ρχ(5)(2 − cos 2η), (84)

L =2c(2 − cos 2η)3ω|χ(3)| sin 2η

√10ρχ(5)(1 − β∗2)

3πσ, (85)

η = arccos( 3

√1 − Q

1 + 2√

1 − Q

) 12

, (86)

Q being given by (13) and

0 < ω2 − k2c∗2 <9πσ(χ(3))2ω2c∗2

10ρχ(5)c2(87)

64


β ≡ v

c∗< 1 . (88)

Equation (81) looks as (15) but the parameters of the kink are changed dueto the condition (88). The formation (81) propagates with a velocity v < c∗.Analogous to the case χ(3) > 0, χ(5) < 0, (8) has also a plane-wave solutionof the type (23) with the parameters (82), (83), (84) and (86).In this case the energy difference ΔW ≡ W − W1 (the self-kink energy) is

ΔW = − 3c3η

2πωc∗2√

30πσρχ(5)(1 − β∗2)

[1 − k2c∗2

ω2

±√

(1 − k2c∗2

ω2)(1 − β∗2) − 27πσc∗2(χ(3))2(1 − 2 cos 2η)

80ρχ(5)c2(2 − cos 2η)2

×(1 +

sin 2η

2η

2 − cos 2η

1 − 2 cos 2η

)].

(89)

(As it was many times mentioned aboveW corresponds to the solitary formationand W1 to the corresponding plane-wave). The energy difference ΔW < 0,i. e. such a kink (anti-kink) formation is energetically more favourable than thecorresponding homogeneous plane-wave (23).For Cs-vapor with atom density N0 = 1017 cm−3 assuming that χ(3) =

−9.2×10−34 esu, χ(5)at = 7.4×10−41 esu λ = 1.06×10−4 cm and η ≈ 0 one

obtains

E0 ≈ 2408η esu , L =0.63

√1 − β∗2

ηcm , (Δq ≈ 0 , ΔΩ ≈ 0) .



L×(cosec2 η cosh2 z − z0 − vt

L − 1)− 1

2(90)

where

q = −k − Δq ,

Δq = ± kβ∗√

1 − β∗2

√ω2

k2c∗2− 1 − 4(1 − β∗2)

k2L2, (91)

Ω = −ω − ΔΩ , ΔΩ = Δqc∗/β∗ , (92)

E0 = ± cot η

2

√3|χ(3)|5ρχ(5)

, (93)

65

Kh. I. Pushkarov

L =2c

ω|χ(3)| sin 2η

√10ρχ(5)(1 − β(∗2)

3πσ, (94)

η = arcsin(1 − Q

) 14

, (95)

Q is given by (13) and the inequalities (87) and (88) must be satisfied.Equation (8) has also a homogeneous solution of the form (67) with the

parameters of (90). The energy difference is

ΔW = − 3c3η

2πωc∗2√

30πσρχ(5)(1 − β∗2)

[1 − k2c∗2

ω2

∓√

(1 − k2c∗2

ω2)(1 − β∗2) − 3πc∗2σ(χ(3))2 sin2 2η

10c2ρχ(5)

− 3πc(∗2)σ(χ(3))2

80c2ρχ(5)

×[5 − 8 cos4 η + 2β∗2(3 sin2 2η − 2 sin2 η)

+sin 2η

2η(3 cos 2η − 4(1 − β∗2) sin2 η)

]].

(96)

The analysis of (96) shows that ΔW can be negative and solitary formationcan be energetically more favourable than (67). The condition (14) is alwayssatisfied.As it is seen from (92) and (91), the frequency of the soliton Ω depends on

the soliton “length” L in z-direction. Hence the effect of self-modulation cantake place. As in the other bell soliton solution above the phase can be changedwith the amplitude.For Cs-vapor:

a) at η ≈ π

2, Δq ≈ ΔΩ ≈ 0, E0 is very small, L ≈ 3.78

√1 − β∗2 cm;

b) ω = kc∗ gives η ≈ 1.31, E0 ≈ 3576 esu, L ≈ 2√

1 − β∗2 cm.



(1 + sech

z − z0 − vt

L)− 1

2

(97)

where q and Ω have the form (82) and (83), respectively,

E0 = ± 34

√|χ(3)|5ρχ(5)

, (98)

66


L =2c

3ω|χ(3)|

√10ρχ(5)(1 − β∗2)

3πσ. (99)

Taking into account that there exists a plane-wave solution of the form (67)with the parameters of (97), we find

ΔW =3c3

8ω3√

30πσρχ(5)(1 − β∗2)

×[ 9π + 4

10π

(ω2

c∗2− k2

)

∓ ω

c∗

√(ω2

c∗2− k2

)(1 − β∗2)

]> 0 .

(100)

Hence, the plane-wave solution (67) is energetically more favourable than(97).

1.2.4. “Inverse” bell soliton solution


(1 − sech

z − z0 − vt

L) 1

2

(101)

where q, Ω, E0 and L are the same is in Subsection 3, the plane-wave solutionhas the form (67) and

ΔW =3c3

8ω3√

30πσρχ(5)(1 − β∗2)

×[9π − 410π

(ω2

c∗2− k2

)

∓ ω

c∗

√(ω2

c∗2− k2

)(1 − β∗2)

]> 0 .

(102)

In this subcase the soliton formation (101) is energetically more favourablethan the plane-wave (67).When the electric field amplitude varies so slowly that (52) is supposed to

take place, Eq. (8) turns into (53) for which the following soliton solutions arefound.

67

Kh. I. Pushkarov

1.2.5. Kink (anti-kink) solution of the form (81)

q, Ω, E0, η and Q are given by (55), (56), (84), (85) (where β∗ = 0), (86)and(13), respectively. Equation (53) has also a homogeneous solution of theform (67) with the parameters of the soliton. In this case

ΔW = − 3c3η

4πωc∗2√

30πσρχ(5)

[β∗2 − k2c∗2

ω2

− 27πc∗2σ(χ(3))2

40c2ρχ(5)

(1 − 2 cos 2η)(2 − cos 2η)2

×(1 +

sin 2η

2η

2 − cos 2η

1 − 2 cos 2η

)].

(103)

According to (103) ΔW can take negative values which means that the solitaryformation can be more favourable energetically.

1.2.6. “Bell” soliton solution of the form (90)

q, Ω, E0, L, η and Q are given by (55), (64), (84), (85) (with β∗ = 0), (86)and (13), respectively.Finally, taking into account that (53) has a homogenous plane-wave solution

of the form (67), for the energy difference ΔW ≡ W − W1 one obtains

ΔW = − 3c3η

4πωc∗2√

30πσρχ(5)

[β∗2 − k2c∗2

ω2

− 3πc∗2σ(χ(3))2(3 − 8 sin4 η)40c2ρχ(5)

×(1 − sin 2η

2η

3(1 + sin4 η) − 14 sin2 η

3 − 8 sin4 η

)].

(104)

2. Conclusions

In this paper a wide variety of solitary formations is shown to exist and propa-gate in nonlinear media for which the polarisation has the form (1). In the mostinteresting cases when χ(3) and χ(5) have opposite signs different kind kinks(anti-kinks), bell (bright) solitons and inverse bell (dark) solitons take place.Some solitary formations can propagate with velocities greater than the phaselight velocity in the medium and without radiation. For the solitary pulsesthe effect of self-modulation can take place (the phase depends on the solitonamplitude). The comparison of the solitary formations and plane-wave ones ismade from an energetical point of view.

68


The analytical solutions obtained for the nonlinear equations deduced as wellas the Lagrangian description of the problem can be used also in the classicaland quantum field theory.

References

1. Kh. Pushkarov. Bulg. J. Phys. 29 (2002).2. Kh. Pushkarov, D. Pushkarov and I. Tomov. Opt. and Quantum Electr. 11 (1979)471.

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