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MNLS solitons
Physics Letters A 339 (2005) 440–445 www.elsevier.com/locate/pla
Calculating adiabatic evolution of the perturbed DNLS/MNLS solitons ✩ Xiang-Jun Chen ∗ , Wa Kun Lam Department of Physics, Jinan University, Guangzhou 510632, PR China Received 1 February 2004; accepted 18 February 2005 Available online 26 February 2005 Communicated by A.R. Bishop
Abstract Adiabatic evolution equations for parameters of the perturbed DNLS/MNLS solitons obtained by recently developed direct perturbation theory [Phys. Rev. E 65 (2002) 066608] are reformulated in simpler forms for practical use. A symbolic computation technique is developed to calculate them, based on the residue theorem. Effects of the third-order dispersion on the MNLS solitons are studied as an example. 2005 Elsevier B.V. All rights reserved. PACS: 05.45.Yv; 52.35.Bj; 42.81.Dp Keywords: DNLS/MNLS solitons; Perturbation; Symbolic computation
1. Introduction It has been well known that exactly integrable nonlinear differential equations support soliton solutions which travel stationarily and collide elastically. It was also known that the physical situations giving rise to exactly solvable equations are highly idealized. Perturbations violating their integrabilities actually exist. ✩ This work was supported by the National Natural Science Foundation of China under Grant No. 10375027. * Corresponding author. E-mail addresses: [email protected] (X.-J. Chen), [email protected] (W.K. Lam).
0375-9601/$ – see front matter 2005 Elsevier B.V. All rights reserved. doi:10.1016/j.physleta.2005.02.030
If these perturbations are small enough, their influence on solitons can still be known analytically by perturbation theories for solitons (see, e.g., [1–3]). In the picture depicted by perturbation theories, the lowest approximation is an adiabatic solution in which the soliton keep its profile unchanged while its parameters such as amplitude, velocity, initial center and initial phase may evolve adiabatically [1,2]. Perturbations can also induce nonadiabatic changes such as radiation emissions. A mathematically complete perturbation theory can obtain not only evolution equations for parameters of solitons analytically but also a formula for calculating the perturbation-induced radiation emission.
X.-J. Chen, W.K. Lam / Physics Letters A 339 (2005) 440–445
The derivative nonlinear Schrödinger (DNLS) equation, iut + uxx + i |u|2 u x = 0, (1) where the subscripts denote partial derivatives, well describes small amplitude nonlinear Alfvén waves in a low-β (the ratio of kinetic to magnetic pressure) plasma, propagating strictly parallel to the ambient magnetic field. Here u, x, and t represent the transverse complex magnetic field, the normalized longitudinal coordinate, and the normalized time, respectively [4–6]. The modified nonlinear Schrödinger (MNLS) equation is 1 iqz + qtt + is |q|2 q t + |q|2 q = 0. (2) 2 It has been suggested that, by including the nonlinear dispersion term (the third term on the left), the MNLS equation is a better model in describing propagation of femtosecond pulses in single mode fibers [7–10] than the nonlinear Schrödinger (NLS, the case when s = 0) equation which is well known to be a good model in describing propagation of picosecond pulses in single mode fibers [11,12]. Here u, z, t, and s denote the normalized electric field, the normalized distance, the normalized time measured in the frame of reference moving with the pulse, and the relative amplitude of the nonlinear dispersion, respectively. We assume s 0 in this Letter because the case for s < 0 can be obtained by a simple transformation t → −t. Both of the DNLS equation and the MNLS equation are exactly integrable. In fact they are connected by a gauge-like transformation (see, e.g., [13]). In this Letter we only consider their solitons with vanishing boundary conditions (i.e., u → 0 as |x| → ∞ and q → 0 as |t| → ∞). The soliton solution for the DNLS was found by the inverse scattering transform technique [14] and the soliton solution for the MNLS can be found by the gauge-like transformation. Perturbation theory for the DNLS/MNLS solitons was first developed by a method based on the Riemann–Hilbert problem [15]. Recently the problem was revisited with a direct perturbation method [16], giving rise to a formula for calculating the perturbation-induced radiation emission and correcting some flaws in Ref. [15] for evolution equations of the initial center and phase. However, evolution equations for the
441
perturbed MNLS solitons are much more complicated than those for the perturbed NLS solitons [1,2]. Problems solved by perturbation theories for DNLS/MNLS solitons [15,16] are limited to some relatively simple perturbations for the MNLS solitons [15–17] so far. In this Letter, in order to make results in Ref. [16] more applicable, we reformulate the evolution equations for the perturbed DNLS/MNLS solitons into simpler expressions and develop a symbolic technique, which is effective for usual perturbations, to calculate them automatically. As an example of the technique, we calculate the evolution of MNLS solitons under influence of the third-order dispersion. Our results approach those for the NLS solitons in the literature as s → 0.
2. Adiabatic evolution of parameters of the perturbed DNLS/MNLS solitons 2.1. Perturbed DNLS solitons The DNLS one-soliton [14] can be rewritten as us (x, t) = −2∆ sin(2β)
cosh(θ − iβ) cosh2 (θ + iβ)
exp(−iϕ), (3)
where θ (x, t) = 2η(x + 4ξ t − x0 ), ξ ϕ(x, t) = 2 θ − 4(ξ 2 + η2 )t + ϕ0 , η
(4)
ξ = ∆2 cos(2β),
(6)
η = ∆2 sin(2β).
(5)
Its amplitude is A = 4∆ sin β.
(7)
Simple calculations yield its energy +∞ E= dx |us |2 = 8β,
(8)
−∞
and the FWHM width 1 w= 2 ln cos β + 1 + cos2 β . ∆ sin(2β)
(9)
In presence of perturbations, the zero on the left-hand side of Eq. (1) should be replaced with the perturbation function ir(u). As β is proportional to the energy, considering evolution of β and ∆, instead of ξ and η
442
X.-J. Chen, W.K. Lam / Physics Letters A 339 (2005) 440–445
in Ref. [16], is more convenient. Evolution equations for parameters of the DNLS solitons in Ref. [16] are reformulated as dβ 1 = dt 4η
and v = s/(1 − 4s 2 ξ ) is its velocity. Similar to the DNLS soliton, we have amplitude A = 4∆ sin β,
∞ dθ R+ (θ, t),
(10)
(20)
energy
−∞
∆ d∆ =i dz 16η
+∞ dθ tanh(θ + iβ)R+ (θ, t),
(11)
and the FWHM width
−∞
i 32η2
∆4 dϕ0 = 3 dt 8η
τ=
∞ dθ R− (θ, t),
(13)
dβ 1 = dz 4η
−∞
−∞
+∞ dθ R+ (θ, z),
R± (θ, z) = u0 (θ ) r0 (θ, z) ± r0 (−θ, z) ,
(14)
u0 (θ ) = us exp(iϕ),
(15)
and the bar stands for complex conjugate in this Letter.
∆ d∆ =i dz 16η
2.2. Perturbed MNLS solitons
qs (t, z) = −2∆ sin(2β)
cosh(θ − iβ) cosh2 (θ + iβ)
exp(−iϕ),
∆4 dϕ0 = 3 dz 8η
where θ (t, z) = 4sη(t − t0 − z/v), ξ z t ϕ(t, z) = 2 θ − 8s 2 ξ 2 + η2 z − + 2 + ϕ0 , η s 2s
η = ∆2 sin(2β),
+∞ cosh(θ + i3β) dθ θ R− (θ, z) cosh(θ + iβ)
−∞
+∞ dθ R− (θ, z),
(25)
−∞
+∞ dθ θ R− (θ, z) −∞
∆2 +i 16η2
(17)
(24)
−∞
i − 64sη2
(16)
ξ = ∆2 cos(2β),
+∞ dθ tanh(θ + iβ)R+ (θ, z),
dt0 ∆2 = dz 32sη3
The MNLS one-soliton is
(23)
−∞
where
r0 (θ, z) = r exp(iϕ),
(22)
of the MNLS soliton. Also, in presence of perturbations, the zero on the right-hand side of Eq. (2) should be replaced with the perturbation function ir(q). Evolution equations for MNLS soliton parameters in Ref. [16] are reformulated as,
+∞ dθ θ R− (θ, t) +∞ cosh(θ − iβ) R− (θ, t), dθ cosh(θ + iβ)
1 ln cos β + 1 + cos2 β , 2 2s∆ sin(2β)
(12)
−∞
∆2 +i 16η2
(21)
−∞
−∞
+∞ ∆2 cosh(θ + i3β) dx0 = R− (θ, t) dθ θ dt cosh(θ + iβ) 16η3 −
+∞ 4β , E= dt|qs |2 = s
+∞ cosh(θ − iβ) dθ R− (θ, z), (26) cosh(θ + iβ)
−∞
where (18)
R± (θ, z) = q0 (θ ) r0 (θ, z) ± r0 (−θ, z) ,
(27)
(19)
q0 (θ ) = qs exp(iϕ),
(28)
r0 (θ, z) = r exp(iϕ).
X.-J. Chen, W.K. Lam / Physics Letters A 339 (2005) 440–445
443
Fig. 1. The path for integrals in Section 3. The crosses are the singularities.
3. A symbolic technique to calculate integrals in evolution equations for soliton parameters In general, integrals in evolution equations for perturbed DNLS/MNLS solitons are rather complicated. Even modern symbolic softwares cannot always tackle them directly. However, taking the perturbed MNLS solitons as an example, most perturbation functions can be expressed as r(q) =
n
rk |q|2 ∂tk q,
where rk (|q|2 ) are complex functions [11,12]. In what follows we will show that for this category of perturbation functions integrals in the evolution equations can be systematically solved by the technique of residue theorem. For perturbation functions in category of Eq. (29), there are only four types of integrands in the evolution equations needed to be tackled. Having been continued to the whole complex plane of θ , they are f (θ ) with f (θ + iπ) = −f (θ ), g(θ ) with g(θ + iπ) = g(θ ), F (θ ) = θf (θ ), G(θ ) = θg(θ ).
∞
f (θ ) dθ = iπ Res f (θ+ ) + Res f (θ− ) , (30)
−∞
and (29)
k=0
(1) (2) (3) (4)
Within the closed path shown in Fig. 1, all of them have and only have a pair of singularities θ± = i(π/2 ± β). Integrations of them on the two vertical line segments are obviously zero. Using the residue theorem on the closed path shown in Fig. 1, we get
∞ F (θ ) dθ −∞
π = −i 2
∞ f (θ ) dθ −∞
+ iπ Res F (θ+ ) + Res F (θ− ) .
(31)
For g(θ ) and G(θ ), by introducing two auxiliary functions, h(θ, ρ) = exp(iρθ )g(θ ),
(32)
and H (θ, ρ) = exp(iρθ )G(θ ),
(33)
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X.-J. Chen, W.K. Lam / Physics Letters A 339 (2005) 440–445
dt0 = α 3s −2 − 12 βη + 4ξ − βξ 2 η−1 dz + 16 13η2 + βηξ − 15ξ 2 + 9βξ 3 η−1 s 2 , (40)
in which ρ > 0, we also have ∞ g(θ ) dθ −∞
= i2π lim
ρ→0+
Res[h(θ+ , ρ)] + Res[h(θ− , ρ)] , (34) 1 − exp(−ρπ)
∞ G(θ ) dθ −∞
Res[h(θ+ , ρ)] + Res[h(θ− , ρ)] = lim −π 2 + ρ→0 2 sinh2 (ρπ/2) Res[H (θ+ , ρ)] + Res[H (θ− , ρ)] . (35) + i2π 1 − exp(−ρπ)
Therefore, despite the fact that these integrals may not be tackled directly, calculation of them comes down to calculation of residues and limitations which can always be done symbolically by modern commercial mathematical softwares.
dϕ0 = α −2s −3 + 12 βη + 4ξ − 12βξ 2 η−1 s −1 dz + 32 −11η2 + βηξ + 3ξ 2 − 3βξ 3 η−1 s + 64 5βη3 − 16η2 ξ + 14βηξ 2 − 16ξ 3 + 9βξ 4 η−1 s 3 . (41) From Eqs. (38)–(41), we see that in presence of the third-order dispersion, within adiabatic approximation, the amplitude, width, and the main velocity v are unchanged. But the third-order dispersion induces linear shifts on t0 and ϕ0 . As s → 0, to keep parameters in the MNLS soliton physically meaningful, ξ and η must be ν µ 1 η→ . ξ→ (42) + 2, 2s 4s 2s The MNLS soliton approach the NLS soliton, qs (t, z) → −2ν sech θ exp(−iϕ),
4. Example: effects of third-order dispersion on the MNLS soliton For the third-order dispersion [11,12] on the MNLS solitons, r(q) = α∂t3 q, we have r0 = α (4sη)3 q0 (θ ) − i3(4sη)2 8sξ − s −1 q0 (θ ) 2 − 3(4sη) 8sξ − s −1 q0 (θ ) 3 + i 8sξ − s −1 q0 (θ ) . (36) With q0 (−θ ) = q0 (θ ),
(37)
we get R+ = 0, R− = 2q¯0 r0 , dβ = 0, dz
(43)
where θ = 2ν(t − tˆ),
tˆ = −2µz + t0 , (44) 2 2 ˆ ϕ = 4µ(t − t ) + ϕ, ˆ ϕˆ = −4 µ + ν z + ϕ0 . (45) Eqs. (38)–(41) become dµ = 0, dz dν = 0, dz dt0 = 4α 3µ2 + ν 2 , dz dϕ0 = 32αµ µ2 − ν 2 . dz They are equivalent to well-known results for solitons in the literature [11,12].
(46) (47) (48) (49) NLS
(38) References
d∆ = 0, dz
(39)
and, using the technique developed in the previous section,
[1] D.J. Kaup, A.C. Newell, Proc. R. Soc. London, Ser. A 361 (1978) 413. [2] V.I. Karpman, E.M. Maslov, Sov. Phys. JETP 46 (1978) 281. [3] D.J. Kaup, Phys. Rev. A 42 (1990) 5689.
X.-J. Chen, W.K. Lam / Physics Letters A 339 (2005) 440–445
[4] A. Rogister, Phys. Fluids 14 (1971) 2733. [5] E. Mjølhus, J. Plasma Phys. 16 (1976) 321. [6] K. Mio, T. Ogino, K. Minami, S. Takeda, J. Phys. Soc. Jpn. 41 (1976) 265. [7] N. Tzoar, M. Jain, Phys. Rev. A 23 (1981) 1266. [8] D. Anderson, M. Lisak, Phys. Rev. A 27 (1983) 1393. [9] K. Ohkuma, Y.H. Ichikawa, Y. Abe, Opt. Lett. 12 (1987) 516. [10] E.V. Doktorov, Eur. Phys. J. B 29 (2002) 227. [11] G.P. Agrawal, Nonlinear Fiber Optics, third ed., Academic Press, New York, 2001.
445
[12] A. Hasegawa, Y. Kodama, Solitons in Optical Communications, Clarendon, Oxford, 1995. [13] Y. Ichikawa, K. Konno, M. Wadati, H. Sanuki, J. Phys. Soc. Jpn. 48 (1980) 279. [14] D.J. Kaup, A.C. Newell, J. Math. Phys. 19 (1978) 798. [15] V.S. Shchesnovich, E.V. Doktorov, Physica D 129 (1998) 115. [16] X.-J. Chen, J. Yang, Phys. Rev. E 65 (2002) 066608. [17] A.A. Afanas’ev, E.V. Doktorov, R.A. Vlasov, V.M. Volkov, Opt. Commun. 153 (1998) 83.
Calculating adiabatic evolution of the perturbed DNLS/MNLS solitons ✩ Xiang-Jun Chen ∗ , Wa Kun Lam Department of Physics, Jinan University, Guangzhou 510632, PR China Received 1 February 2004; accepted 18 February 2005 Available online 26 February 2005 Communicated by A.R. Bishop
Abstract Adiabatic evolution equations for parameters of the perturbed DNLS/MNLS solitons obtained by recently developed direct perturbation theory [Phys. Rev. E 65 (2002) 066608] are reformulated in simpler forms for practical use. A symbolic computation technique is developed to calculate them, based on the residue theorem. Effects of the third-order dispersion on the MNLS solitons are studied as an example. 2005 Elsevier B.V. All rights reserved. PACS: 05.45.Yv; 52.35.Bj; 42.81.Dp Keywords: DNLS/MNLS solitons; Perturbation; Symbolic computation
1. Introduction It has been well known that exactly integrable nonlinear differential equations support soliton solutions which travel stationarily and collide elastically. It was also known that the physical situations giving rise to exactly solvable equations are highly idealized. Perturbations violating their integrabilities actually exist. ✩ This work was supported by the National Natural Science Foundation of China under Grant No. 10375027. * Corresponding author. E-mail addresses: [email protected] (X.-J. Chen), [email protected] (W.K. Lam).
0375-9601/$ – see front matter 2005 Elsevier B.V. All rights reserved. doi:10.1016/j.physleta.2005.02.030
If these perturbations are small enough, their influence on solitons can still be known analytically by perturbation theories for solitons (see, e.g., [1–3]). In the picture depicted by perturbation theories, the lowest approximation is an adiabatic solution in which the soliton keep its profile unchanged while its parameters such as amplitude, velocity, initial center and initial phase may evolve adiabatically [1,2]. Perturbations can also induce nonadiabatic changes such as radiation emissions. A mathematically complete perturbation theory can obtain not only evolution equations for parameters of solitons analytically but also a formula for calculating the perturbation-induced radiation emission.
X.-J. Chen, W.K. Lam / Physics Letters A 339 (2005) 440–445
The derivative nonlinear Schrödinger (DNLS) equation, iut + uxx + i |u|2 u x = 0, (1) where the subscripts denote partial derivatives, well describes small amplitude nonlinear Alfvén waves in a low-β (the ratio of kinetic to magnetic pressure) plasma, propagating strictly parallel to the ambient magnetic field. Here u, x, and t represent the transverse complex magnetic field, the normalized longitudinal coordinate, and the normalized time, respectively [4–6]. The modified nonlinear Schrödinger (MNLS) equation is 1 iqz + qtt + is |q|2 q t + |q|2 q = 0. (2) 2 It has been suggested that, by including the nonlinear dispersion term (the third term on the left), the MNLS equation is a better model in describing propagation of femtosecond pulses in single mode fibers [7–10] than the nonlinear Schrödinger (NLS, the case when s = 0) equation which is well known to be a good model in describing propagation of picosecond pulses in single mode fibers [11,12]. Here u, z, t, and s denote the normalized electric field, the normalized distance, the normalized time measured in the frame of reference moving with the pulse, and the relative amplitude of the nonlinear dispersion, respectively. We assume s 0 in this Letter because the case for s < 0 can be obtained by a simple transformation t → −t. Both of the DNLS equation and the MNLS equation are exactly integrable. In fact they are connected by a gauge-like transformation (see, e.g., [13]). In this Letter we only consider their solitons with vanishing boundary conditions (i.e., u → 0 as |x| → ∞ and q → 0 as |t| → ∞). The soliton solution for the DNLS was found by the inverse scattering transform technique [14] and the soliton solution for the MNLS can be found by the gauge-like transformation. Perturbation theory for the DNLS/MNLS solitons was first developed by a method based on the Riemann–Hilbert problem [15]. Recently the problem was revisited with a direct perturbation method [16], giving rise to a formula for calculating the perturbation-induced radiation emission and correcting some flaws in Ref. [15] for evolution equations of the initial center and phase. However, evolution equations for the
441
perturbed MNLS solitons are much more complicated than those for the perturbed NLS solitons [1,2]. Problems solved by perturbation theories for DNLS/MNLS solitons [15,16] are limited to some relatively simple perturbations for the MNLS solitons [15–17] so far. In this Letter, in order to make results in Ref. [16] more applicable, we reformulate the evolution equations for the perturbed DNLS/MNLS solitons into simpler expressions and develop a symbolic technique, which is effective for usual perturbations, to calculate them automatically. As an example of the technique, we calculate the evolution of MNLS solitons under influence of the third-order dispersion. Our results approach those for the NLS solitons in the literature as s → 0.
2. Adiabatic evolution of parameters of the perturbed DNLS/MNLS solitons 2.1. Perturbed DNLS solitons The DNLS one-soliton [14] can be rewritten as us (x, t) = −2∆ sin(2β)
cosh(θ − iβ) cosh2 (θ + iβ)
exp(−iϕ), (3)
where θ (x, t) = 2η(x + 4ξ t − x0 ), ξ ϕ(x, t) = 2 θ − 4(ξ 2 + η2 )t + ϕ0 , η
(4)
ξ = ∆2 cos(2β),
(6)
η = ∆2 sin(2β).
(5)
Its amplitude is A = 4∆ sin β.
(7)
Simple calculations yield its energy +∞ E= dx |us |2 = 8β,
(8)
−∞
and the FWHM width 1 w= 2 ln cos β + 1 + cos2 β . ∆ sin(2β)
(9)
In presence of perturbations, the zero on the left-hand side of Eq. (1) should be replaced with the perturbation function ir(u). As β is proportional to the energy, considering evolution of β and ∆, instead of ξ and η
442
X.-J. Chen, W.K. Lam / Physics Letters A 339 (2005) 440–445
in Ref. [16], is more convenient. Evolution equations for parameters of the DNLS solitons in Ref. [16] are reformulated as dβ 1 = dt 4η
and v = s/(1 − 4s 2 ξ ) is its velocity. Similar to the DNLS soliton, we have amplitude A = 4∆ sin β,
∞ dθ R+ (θ, t),
(10)
(20)
energy
−∞
∆ d∆ =i dz 16η
+∞ dθ tanh(θ + iβ)R+ (θ, t),
(11)
and the FWHM width
−∞
i 32η2
∆4 dϕ0 = 3 dt 8η
τ=
∞ dθ R− (θ, t),
(13)
dβ 1 = dz 4η
−∞
−∞
+∞ dθ R+ (θ, z),
R± (θ, z) = u0 (θ ) r0 (θ, z) ± r0 (−θ, z) ,
(14)
u0 (θ ) = us exp(iϕ),
(15)
and the bar stands for complex conjugate in this Letter.
∆ d∆ =i dz 16η
2.2. Perturbed MNLS solitons
qs (t, z) = −2∆ sin(2β)
cosh(θ − iβ) cosh2 (θ + iβ)
exp(−iϕ),
∆4 dϕ0 = 3 dz 8η
where θ (t, z) = 4sη(t − t0 − z/v), ξ z t ϕ(t, z) = 2 θ − 8s 2 ξ 2 + η2 z − + 2 + ϕ0 , η s 2s
η = ∆2 sin(2β),
+∞ cosh(θ + i3β) dθ θ R− (θ, z) cosh(θ + iβ)
−∞
+∞ dθ R− (θ, z),
(25)
−∞
+∞ dθ θ R− (θ, z) −∞
∆2 +i 16η2
(17)
(24)
−∞
i − 64sη2
(16)
ξ = ∆2 cos(2β),
+∞ dθ tanh(θ + iβ)R+ (θ, z),
dt0 ∆2 = dz 32sη3
The MNLS one-soliton is
(23)
−∞
where
r0 (θ, z) = r exp(iϕ),
(22)
of the MNLS soliton. Also, in presence of perturbations, the zero on the right-hand side of Eq. (2) should be replaced with the perturbation function ir(q). Evolution equations for MNLS soliton parameters in Ref. [16] are reformulated as,
+∞ dθ θ R− (θ, t) +∞ cosh(θ − iβ) R− (θ, t), dθ cosh(θ + iβ)
1 ln cos β + 1 + cos2 β , 2 2s∆ sin(2β)
(12)
−∞
∆2 +i 16η2
(21)
−∞
−∞
+∞ ∆2 cosh(θ + i3β) dx0 = R− (θ, t) dθ θ dt cosh(θ + iβ) 16η3 −
+∞ 4β , E= dt|qs |2 = s
+∞ cosh(θ − iβ) dθ R− (θ, z), (26) cosh(θ + iβ)
−∞
where (18)
R± (θ, z) = q0 (θ ) r0 (θ, z) ± r0 (−θ, z) ,
(27)
(19)
q0 (θ ) = qs exp(iϕ),
(28)
r0 (θ, z) = r exp(iϕ).
X.-J. Chen, W.K. Lam / Physics Letters A 339 (2005) 440–445
443
Fig. 1. The path for integrals in Section 3. The crosses are the singularities.
3. A symbolic technique to calculate integrals in evolution equations for soliton parameters In general, integrals in evolution equations for perturbed DNLS/MNLS solitons are rather complicated. Even modern symbolic softwares cannot always tackle them directly. However, taking the perturbed MNLS solitons as an example, most perturbation functions can be expressed as r(q) =
n
rk |q|2 ∂tk q,
where rk (|q|2 ) are complex functions [11,12]. In what follows we will show that for this category of perturbation functions integrals in the evolution equations can be systematically solved by the technique of residue theorem. For perturbation functions in category of Eq. (29), there are only four types of integrands in the evolution equations needed to be tackled. Having been continued to the whole complex plane of θ , they are f (θ ) with f (θ + iπ) = −f (θ ), g(θ ) with g(θ + iπ) = g(θ ), F (θ ) = θf (θ ), G(θ ) = θg(θ ).
∞
f (θ ) dθ = iπ Res f (θ+ ) + Res f (θ− ) , (30)
−∞
and (29)
k=0
(1) (2) (3) (4)
Within the closed path shown in Fig. 1, all of them have and only have a pair of singularities θ± = i(π/2 ± β). Integrations of them on the two vertical line segments are obviously zero. Using the residue theorem on the closed path shown in Fig. 1, we get
∞ F (θ ) dθ −∞
π = −i 2
∞ f (θ ) dθ −∞
+ iπ Res F (θ+ ) + Res F (θ− ) .
(31)
For g(θ ) and G(θ ), by introducing two auxiliary functions, h(θ, ρ) = exp(iρθ )g(θ ),
(32)
and H (θ, ρ) = exp(iρθ )G(θ ),
(33)
444
X.-J. Chen, W.K. Lam / Physics Letters A 339 (2005) 440–445
dt0 = α 3s −2 − 12 βη + 4ξ − βξ 2 η−1 dz + 16 13η2 + βηξ − 15ξ 2 + 9βξ 3 η−1 s 2 , (40)
in which ρ > 0, we also have ∞ g(θ ) dθ −∞
= i2π lim
ρ→0+
Res[h(θ+ , ρ)] + Res[h(θ− , ρ)] , (34) 1 − exp(−ρπ)
∞ G(θ ) dθ −∞
Res[h(θ+ , ρ)] + Res[h(θ− , ρ)] = lim −π 2 + ρ→0 2 sinh2 (ρπ/2) Res[H (θ+ , ρ)] + Res[H (θ− , ρ)] . (35) + i2π 1 − exp(−ρπ)
Therefore, despite the fact that these integrals may not be tackled directly, calculation of them comes down to calculation of residues and limitations which can always be done symbolically by modern commercial mathematical softwares.
dϕ0 = α −2s −3 + 12 βη + 4ξ − 12βξ 2 η−1 s −1 dz + 32 −11η2 + βηξ + 3ξ 2 − 3βξ 3 η−1 s + 64 5βη3 − 16η2 ξ + 14βηξ 2 − 16ξ 3 + 9βξ 4 η−1 s 3 . (41) From Eqs. (38)–(41), we see that in presence of the third-order dispersion, within adiabatic approximation, the amplitude, width, and the main velocity v are unchanged. But the third-order dispersion induces linear shifts on t0 and ϕ0 . As s → 0, to keep parameters in the MNLS soliton physically meaningful, ξ and η must be ν µ 1 η→ . ξ→ (42) + 2, 2s 4s 2s The MNLS soliton approach the NLS soliton, qs (t, z) → −2ν sech θ exp(−iϕ),
4. Example: effects of third-order dispersion on the MNLS soliton For the third-order dispersion [11,12] on the MNLS solitons, r(q) = α∂t3 q, we have r0 = α (4sη)3 q0 (θ ) − i3(4sη)2 8sξ − s −1 q0 (θ ) 2 − 3(4sη) 8sξ − s −1 q0 (θ ) 3 + i 8sξ − s −1 q0 (θ ) . (36) With q0 (−θ ) = q0 (θ ),
(37)
we get R+ = 0, R− = 2q¯0 r0 , dβ = 0, dz
(43)
where θ = 2ν(t − tˆ),
tˆ = −2µz + t0 , (44) 2 2 ˆ ϕ = 4µ(t − t ) + ϕ, ˆ ϕˆ = −4 µ + ν z + ϕ0 . (45) Eqs. (38)–(41) become dµ = 0, dz dν = 0, dz dt0 = 4α 3µ2 + ν 2 , dz dϕ0 = 32αµ µ2 − ν 2 . dz They are equivalent to well-known results for solitons in the literature [11,12].
(46) (47) (48) (49) NLS
(38) References
d∆ = 0, dz
(39)
and, using the technique developed in the previous section,
[1] D.J. Kaup, A.C. Newell, Proc. R. Soc. London, Ser. A 361 (1978) 413. [2] V.I. Karpman, E.M. Maslov, Sov. Phys. JETP 46 (1978) 281. [3] D.J. Kaup, Phys. Rev. A 42 (1990) 5689.
X.-J. Chen, W.K. Lam / Physics Letters A 339 (2005) 440–445
[4] A. Rogister, Phys. Fluids 14 (1971) 2733. [5] E. Mjølhus, J. Plasma Phys. 16 (1976) 321. [6] K. Mio, T. Ogino, K. Minami, S. Takeda, J. Phys. Soc. Jpn. 41 (1976) 265. [7] N. Tzoar, M. Jain, Phys. Rev. A 23 (1981) 1266. [8] D. Anderson, M. Lisak, Phys. Rev. A 27 (1983) 1393. [9] K. Ohkuma, Y.H. Ichikawa, Y. Abe, Opt. Lett. 12 (1987) 516. [10] E.V. Doktorov, Eur. Phys. J. B 29 (2002) 227. [11] G.P. Agrawal, Nonlinear Fiber Optics, third ed., Academic Press, New York, 2001.
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[12] A. Hasegawa, Y. Kodama, Solitons in Optical Communications, Clarendon, Oxford, 1995. [13] Y. Ichikawa, K. Konno, M. Wadati, H. Sanuki, J. Phys. Soc. Jpn. 48 (1980) 279. [14] D.J. Kaup, A.C. Newell, J. Math. Phys. 19 (1978) 798. [15] V.S. Shchesnovich, E.V. Doktorov, Physica D 129 (1998) 115. [16] X.-J. Chen, J. Yang, Phys. Rev. E 65 (2002) 066608. [17] A.A. Afanas’ev, E.V. Doktorov, R.A. Vlasov, V.M. Volkov, Opt. Commun. 153 (1998) 83.