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Perturbation theory and optical soliton cooling with anti-cubic nonlinearity
Accepted Manuscript Title: PERTURBATION THEORY AND OPTICAL SOLITON COOLING WITH ANTI-CUBIC NONLINEARITY Author: Anjan Biswas Qin Zhou Malik Zaka Ullah Mir Asma Seithuti P. Moshokoa Milivoj Belic PII: DOI: Reference:
S0030-4026(17)30596-X http://dx.doi.org/doi:10.1016/j.ijleo.2017.05.060 IJLEO 59206
To appear in: Received date: Accepted date:
21-2-2017 17-5-2017
Please cite this article as: Anjan Biswas, Qin Zhou, Malik Zaka Ullah, Mir Asma, Seithuti P. Moshokoa, Milivoj Belic, PERTURBATION THEORY AND OPTICAL SOLITON COOLING WITH ANTI-CUBIC NONLINEARITY, (2017), http://dx.doi.org/10.1016/j.ijleo.2017.05.060 This is a PDF file of an unedited manuscript that has been accepted for publication. As a service to our customers we are providing this early version of the manuscript. The manuscript will undergo copyediting, typesetting, and review of the resulting proof before it is published in its final form. Please note that during the production process errors may be discovered which could affect the content, and all legal disclaimers that apply to the journal pertain.
cr
PERTURBATION THEORY AND OPTICAL SOLITON COOLING WITH ANTI-CUBIC NONLINEARITY 5
us
Anjan Biswas 1, 2 , Qin Zhou 3 , Malik Zaka Ullah 2 , Mir Asma 4 , Seithuti P. Moshokoa 1 & Milivoj Belic 1
an
Department of Mathematics and Statistics, Tshwane University of Technology, Pretoria-0008, South Africa 2
M
Operator Theory and Applications Research Group, Department of Mathematics, Faculty of Science, King Abdulaziz University, PO Box-80203, Jeddah-21589, Saudi Arabia 3
Institute of Mathematical Sciences, Faculty of Science, University of Malaya, 50601 Kuala Lumpur, Malaysia 5
te
4
d
School of Electronics and Information Engineering, Wuhan Donghu University, Wuhan 430212, People’s Republic of China
Science Program, Texas A & M University at Qatar, PO Box 23874, Doha, Qatar
Ac ce p
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65
ip t
*Manuscript
Abstract
Soliton perturbation theory is applied to obtain the adiabatic variation of its parameters and slow change in velocity. The dynamical system leads to a stable fixed point to which the soliton amplitude and frequency gets locked into for a stable propagation down the fibers with anti-cubic nonlinearity.
Keywords: solitons; perturbation; adiabaticity.
1
Page 1 of 6
1
INTRODUCTION
2
an
us
cr
ip t
Optical soliton perturbation is inevitable in long distance fiber optic communication technology [1-10]. It is therefore imperative to handle it in a scientific manner. Soliton perturbation theory has been addressed in the past for various forms of nonlinear fibers. These are Kerr law, power law, parabolic law, dual-power law and log-law nonlinearities [1, 2]. This paper addresses the issue for a fairly new form of nonlinearity that has been lately proposed. It is the anti-cubic law that first appeared during 2003 [4]. The exact 1-soliton solution for the nonlinear Schr¨odinger’s equation (NLSE) with this law has already been reported [7, 10]. This will lead to the adiabatic variation of soliton parameters such as its energy and frequency when the perturbation terms are turned on. The dynamical system with these two parameters will lead to a stable fixed point to which these parameters gets locked into during pulse propagation down these fibers. The soliton velocity also undergoes a slow change, with perturbation, that will be presented in this paper as well.
MATHEMATICAL ANALYSIS
M
The dimensionless form of NLSE with anti-cubic nonlinearity is given by [4, 7, 10] −4 2 4 iqt + aqxx + b1 |q| + b2 |q| + b3 |q| q = 0.
(1)
te
d
In (1) q(x, t) is the complex-valued wave profile where x and t are spatial and temporal variables respectively that are independent of each other. The first term is the temporal evolution. The coefficient of a is the group velocity dispersion (GVD). The last three terms form the nonlinearity for NLSE. The coefficient of b1 gives the anti-cubic effect. For b1 = 0, (1) reduces to NLSE with parabolic law nonlinearity that has been extensively studied in the past. The exact 1-soliton solution to (1) is given by [7, 10]: p (2) q(x, t) = A sech [B (x − vt)]ei(−κx+ωt+θ0 )
Ac ce p
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65
where A is the amplitude of the soliton, B is its inverse width and v s the speed at which the soliton is traveling. From the phase component, κ is the soliton frequency, ω is the soliton wave number and θ0 is the phase constant. These parameters are connected as follows: s 3 (ω + aκ2 ) A= , (3) b3
B=2
r
ω + aκ2 a
(4)
and v = −2aκ.
(5)
a ω + aκ2 > 0,
(6)
b3 ω + aκ2 > 0.
(7)
This soliton solution is valid with the constraints
and
2
Page 2 of 6
2.1
CONSERVATION LAWS
ip t
The first two conservation laws for NLSE with anti-cubic nonlinearity as reported earlier are [7, 10]: Z ∞ πA2 E= |q|2 dx = , B −∞
M = ia
Z
∞
(qqx∗ − q ∗ qx ) dx = aκ
(8)
(9)
us
−∞
πA2 = aκE, B
cr
and
which represents the energy (power) and linear momentum of the soliton respectively. These conserved quantities are evaluated using the 1-soliton solution as given by (2).
PERTURBATION TERMS
an
2.2
(10)
M
The perturbed NLSE with anti-cubic nonlinearity is given by [5, 8] iqt + aqxx + b1 |q|−4 + b2 |q|2 + b3 |q|4 q = iǫR,
and
te
d
where R represents the perturbation terms and ǫ is the perturbation parameter that arises due to quasi-monochromaticity [6]. In presence of perturbation terms soliton parameter such as its energy and frequency undergo adiabatic deformation and they are given by: Z ∞ dE =ǫ (q ∗ R + qR∗ ) dx (11) dt −∞ Z ∞ Z ∞ dκ ǫ ∗ ∗ ∗ ∗ i (qx R − qx R ) dx − aκ (q R + qR ) dx . = dt E −∞ −∞
Ac ce p
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65
(12)
Finally, the change of soliton velocity is given by v = −2aκ −
3
ǫ E
Z
∞
x (q ∗ R + qR∗ ) dx.
(13)
−∞
SOLITON PERTURBATION THEORY
This section will look into specific perturbation terms that arises in the context of soliton propagation through optical fibers. The soliton perturbation theory (SPT) will lead to the adiabatic variations of the soliton energy and linear momentum.
3.1
PERTURBATION TERMS
The perturbation terms that will be studied in this paper are [5, 8]: R = δ |q|2m q + αqx + βqxx − γqxxx + λ |q|2 q + θ |q|2 q + ρ |qx |2 q x x 2 2 2 ∗ ∗ 2 ∗ q − iξ q qx x − iηqx q − iζq q xx − iµ |q| x Z x 2 |q| ds. − iχqxxxx − iψqxxxxx + (σ1 q + σ2 qx )
(14)
−∞
3
Page 3 of 6
an
us
cr
ip t
In (14), δ represents multi-photon absorption and α is the inter-modal dispersion that occurs when group velocity of light propagating in a multimode fiber depends not only on the optical frequency (also known as chromatic dispersion) but also on the propagation mode involved. The coefficient of β is the bandpass filter. The coefficient is the third order dispersion γ is considered when GVD is low. Also λ is the self-steepening term that avoids the formation of shock waves and θ represents nonlinear dispersion. The nonlinear dissipation is given by ρ. The perturbations due to quasi-solitons are represented by ξ, η and ζ. The fourth and sixth order dispersions are respectively represented by χ and ψ. Finally, the saturable amplifiers are given by σj for j = 1, 2. With these perturbation terms, the perturbed NLSE, given by (10), changes to: −4 2 4 iqt + aqxx + b1 |q| + b2 |q| + b3 |q| q h 2m 2 2 2 = iǫ δ |q| q + αqx + βqxx − γqxxx + λ |q| q + θ |q| q + ρ |qx | q x x 2 q − iξ q 2 qx∗ x − iηqx2 q ∗ − iζq ∗ q 2 xx − iµ |q| x Z x 2 |q| ds (15) − iχqxxxx − iψqxxxxx + (σ1 q + σ2 qx ) −∞
ADIABATIC PARAMETER DYNAMICS
M
3.2
and
te
d
With the perturbed NLSE given by (15), the adiabatic variation of the soliton energy and frequency as indicated by (11) and (12) are given by [5, 8] " ρA4 Γ 12 dE 4δA2m+2 Γ m+1 πβA2 2 B 2 + 8κ2 + B 2 + 12κ2 = ǫ − m dt mB 4B 3B Γ 2 2 4 4 Z ∞ σ1 π A 2σ2 A −1 + − sechs tanh s tan (sinh s) ds , (16) B2 B −∞
Ac ce p
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65
Z ∞ ǫA2 dκ 2 −1 6βκπB − 2µA B + 3σ2 κ sechs tanh s tan (sinh s) ds . = dt 3aE −∞
The velocity change by virtue of (13) takes the form: A2 3γ 2κA2 v = −2aκ − ǫ α + B 2 + κ2 + (3λ + 2θ) + (ξ − η − 2ζ) 8 π π Z 2σ1 A2 ∞ χκ σ2 πA2 B 2 + 8κ2 − + s sechs tan−1 (sinh s) ds + π 2B 2 −∞ 3ψκB − 11 3 + 4κ2 B 4 + 32 1 + 3κ2 B 2 + 128κ4 . 64
(17)
(18)
In order to evaluate the integrals in (11)-(13), the 1-soliton solution given by (2) is utilized.
¯ κ ¯ 0) The dynamical system given by (16) and (17) for µ = ρ = 0 has a stable fixed point given by (A, ¯ ) = (A, ¯ κ ¯ 0) where A¯ and B ¯ are related through the equation [5] or (B, ¯ ) = (B, m+1 Γ 21 2m Γ 2 − mπβB 3 + 4σ1 mπ 2 A2 − 8σ2 mA2 BJ = 0, (19) 16δA B Γ m 2
with
J=
Z
∞
sechs tanh s tan−1 (sinh s) ds.
(20)
−∞
4
Page 4 of 6
4
ip t
This shows that the perturbed soliton moves at the speed of light with a fixed value of the amplitude and width that are given by the solution of (19). This phenomena is known as optical soliton cooling [6].
CONCLUSIONS
an
us
cr
This paper obtained the adiabatic variation of soliton power (energy) and frequency in presence of perturbation terms for anti-cubic nonlinearity. The slow change in soliton velocity is also presented. The corresponding dynamical system led to a stable fixed point and hence the phenomenon of soliton cooling is enlightened for such nonlinearity. These results for anti-cubic nonlinearity are being reported for the first time in this paper. They carry lot of future prospects. The analysis can be extended to other type of nonlinearities that have not been covered. The results of this paper leads to the formulation of quasi-particle theory for supressing the intra-channel soliton-soliton interaction. The outcome of those research is currently awaited, but will be soon disseminated. ACKNOWLEDGEMENTS
te
d
M
The work of the second author (QZ) was supported by the National Science Foundation for Young Scientists of Wuhan Donghu University. The fifth author (SPM) would like to thank the research support provided by the Department of Mathematics and Statistics at Tshwane University of Technology and the support from the South African National Foundation under Grant Number 92052 IRF1202210126. The research work of sixth author (MB) was supported by Qatar National Research Fund (QNRF) under the grant number NPRP 6-021-1-005. The authors also declare that there is no conflict of interest.
Ac ce p
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65
5
Page 5 of 6
References
ip t
[1] N. N. Akhmediev, A. Ankiewicz & R. Grimshaw. “Hamiltonian-versus energy diagrams in soliton theory”. Physical Review E. Volume 59 Issue 5, 6088-6096. (1999). [2] A. Biswas & S. Konar. Introduction to non-Kerr law optical solitons. CRC Press. Boca Raton, FL. (2006).
cr
[3] M. Ekici, M. Mirzazadeh, A. Sonmezoglu, M. Z. Ullah, Q. Zhou, H. Triki, S. P. Moshokoa, & A. Biswas. “Optical solitons with anti-cubic nonlinearity by extended trial equation method”. To appear in Optik.
us
[4] R. Fedele, H. Schamel, V. I. Karpman & P. K. Shukla. “Envelope solitons of nonlinear Schr¨odinger equation with an anti-cubic nonlinearity”. Journal of Physics A. Volume 36, 1169-1173. (2003).
an
[5] L. Girgis, D. Milovic, T. Hayat, O. M. Aldossary & A. Biswas. “Optical soliton perturbation with log law nonlinearity”. Optica Applicata. Volume 32, Number 3, 447–454. (2012). [6] A. Hasegawa & Y. Kodama. Solitons in optical communications. Oxford University Press. UK. (1995).
M
[7] A. J. M. Jawad, M. Mirzazadeh, Q. Zhou & A. Biswas. “Optical solitons with anti-cubic nonlinearity using three integration schemes”. Submitted. [8] R. Kohl, A. Biswas, D. Milovic & E. Zerrad. “Optical soliton perturbation in a non-Kerr law media”. Optics and Laser Technology. Volume 40 Issue 4, 647-662. (2008).
te
d
[9] Z. Li, L. Li, H. Tian & G. Zhou. “New types of solitary wave solutions for the higher order nonlinear Schr¨odinger equation”. Physical Review Letters. Volume 84, Number 18, 4096–4099. (2000). [10] H. Triki, A. H. Kara, A. Biswas, S. P. Moshokoa & M. Belic. “Optical solitons and conservation laws with anti-cubic nonlinearity”. Optik. Volume 127, Issue 24, 12056-12062. (2016).
Ac ce p
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65
6
Page 6 of 6
S0030-4026(17)30596-X http://dx.doi.org/doi:10.1016/j.ijleo.2017.05.060 IJLEO 59206
To appear in: Received date: Accepted date:
21-2-2017 17-5-2017
Please cite this article as: Anjan Biswas, Qin Zhou, Malik Zaka Ullah, Mir Asma, Seithuti P. Moshokoa, Milivoj Belic, PERTURBATION THEORY AND OPTICAL SOLITON COOLING WITH ANTI-CUBIC NONLINEARITY, (2017), http://dx.doi.org/10.1016/j.ijleo.2017.05.060 This is a PDF file of an unedited manuscript that has been accepted for publication. As a service to our customers we are providing this early version of the manuscript. The manuscript will undergo copyediting, typesetting, and review of the resulting proof before it is published in its final form. Please note that during the production process errors may be discovered which could affect the content, and all legal disclaimers that apply to the journal pertain.
cr
PERTURBATION THEORY AND OPTICAL SOLITON COOLING WITH ANTI-CUBIC NONLINEARITY 5
us
Anjan Biswas 1, 2 , Qin Zhou 3 , Malik Zaka Ullah 2 , Mir Asma 4 , Seithuti P. Moshokoa 1 & Milivoj Belic 1
an
Department of Mathematics and Statistics, Tshwane University of Technology, Pretoria-0008, South Africa 2
M
Operator Theory and Applications Research Group, Department of Mathematics, Faculty of Science, King Abdulaziz University, PO Box-80203, Jeddah-21589, Saudi Arabia 3
Institute of Mathematical Sciences, Faculty of Science, University of Malaya, 50601 Kuala Lumpur, Malaysia 5
te
4
d
School of Electronics and Information Engineering, Wuhan Donghu University, Wuhan 430212, People’s Republic of China
Science Program, Texas A & M University at Qatar, PO Box 23874, Doha, Qatar
Ac ce p
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65
ip t
*Manuscript
Abstract
Soliton perturbation theory is applied to obtain the adiabatic variation of its parameters and slow change in velocity. The dynamical system leads to a stable fixed point to which the soliton amplitude and frequency gets locked into for a stable propagation down the fibers with anti-cubic nonlinearity.
Keywords: solitons; perturbation; adiabaticity.
1
Page 1 of 6
1
INTRODUCTION
2
an
us
cr
ip t
Optical soliton perturbation is inevitable in long distance fiber optic communication technology [1-10]. It is therefore imperative to handle it in a scientific manner. Soliton perturbation theory has been addressed in the past for various forms of nonlinear fibers. These are Kerr law, power law, parabolic law, dual-power law and log-law nonlinearities [1, 2]. This paper addresses the issue for a fairly new form of nonlinearity that has been lately proposed. It is the anti-cubic law that first appeared during 2003 [4]. The exact 1-soliton solution for the nonlinear Schr¨odinger’s equation (NLSE) with this law has already been reported [7, 10]. This will lead to the adiabatic variation of soliton parameters such as its energy and frequency when the perturbation terms are turned on. The dynamical system with these two parameters will lead to a stable fixed point to which these parameters gets locked into during pulse propagation down these fibers. The soliton velocity also undergoes a slow change, with perturbation, that will be presented in this paper as well.
MATHEMATICAL ANALYSIS
M
The dimensionless form of NLSE with anti-cubic nonlinearity is given by [4, 7, 10] −4 2 4 iqt + aqxx + b1 |q| + b2 |q| + b3 |q| q = 0.
(1)
te
d
In (1) q(x, t) is the complex-valued wave profile where x and t are spatial and temporal variables respectively that are independent of each other. The first term is the temporal evolution. The coefficient of a is the group velocity dispersion (GVD). The last three terms form the nonlinearity for NLSE. The coefficient of b1 gives the anti-cubic effect. For b1 = 0, (1) reduces to NLSE with parabolic law nonlinearity that has been extensively studied in the past. The exact 1-soliton solution to (1) is given by [7, 10]: p (2) q(x, t) = A sech [B (x − vt)]ei(−κx+ωt+θ0 )
Ac ce p
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65
where A is the amplitude of the soliton, B is its inverse width and v s the speed at which the soliton is traveling. From the phase component, κ is the soliton frequency, ω is the soliton wave number and θ0 is the phase constant. These parameters are connected as follows: s 3 (ω + aκ2 ) A= , (3) b3
B=2
r
ω + aκ2 a
(4)
and v = −2aκ.
(5)
a ω + aκ2 > 0,
(6)
b3 ω + aκ2 > 0.
(7)
This soliton solution is valid with the constraints
and
2
Page 2 of 6
2.1
CONSERVATION LAWS
ip t
The first two conservation laws for NLSE with anti-cubic nonlinearity as reported earlier are [7, 10]: Z ∞ πA2 E= |q|2 dx = , B −∞
M = ia
Z
∞
(qqx∗ − q ∗ qx ) dx = aκ
(8)
(9)
us
−∞
πA2 = aκE, B
cr
and
which represents the energy (power) and linear momentum of the soliton respectively. These conserved quantities are evaluated using the 1-soliton solution as given by (2).
PERTURBATION TERMS
an
2.2
(10)
M
The perturbed NLSE with anti-cubic nonlinearity is given by [5, 8] iqt + aqxx + b1 |q|−4 + b2 |q|2 + b3 |q|4 q = iǫR,
and
te
d
where R represents the perturbation terms and ǫ is the perturbation parameter that arises due to quasi-monochromaticity [6]. In presence of perturbation terms soliton parameter such as its energy and frequency undergo adiabatic deformation and they are given by: Z ∞ dE =ǫ (q ∗ R + qR∗ ) dx (11) dt −∞ Z ∞ Z ∞ dκ ǫ ∗ ∗ ∗ ∗ i (qx R − qx R ) dx − aκ (q R + qR ) dx . = dt E −∞ −∞
Ac ce p
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65
(12)
Finally, the change of soliton velocity is given by v = −2aκ −
3
ǫ E
Z
∞
x (q ∗ R + qR∗ ) dx.
(13)
−∞
SOLITON PERTURBATION THEORY
This section will look into specific perturbation terms that arises in the context of soliton propagation through optical fibers. The soliton perturbation theory (SPT) will lead to the adiabatic variations of the soliton energy and linear momentum.
3.1
PERTURBATION TERMS
The perturbation terms that will be studied in this paper are [5, 8]: R = δ |q|2m q + αqx + βqxx − γqxxx + λ |q|2 q + θ |q|2 q + ρ |qx |2 q x x 2 2 2 ∗ ∗ 2 ∗ q − iξ q qx x − iηqx q − iζq q xx − iµ |q| x Z x 2 |q| ds. − iχqxxxx − iψqxxxxx + (σ1 q + σ2 qx )
(14)
−∞
3
Page 3 of 6
an
us
cr
ip t
In (14), δ represents multi-photon absorption and α is the inter-modal dispersion that occurs when group velocity of light propagating in a multimode fiber depends not only on the optical frequency (also known as chromatic dispersion) but also on the propagation mode involved. The coefficient of β is the bandpass filter. The coefficient is the third order dispersion γ is considered when GVD is low. Also λ is the self-steepening term that avoids the formation of shock waves and θ represents nonlinear dispersion. The nonlinear dissipation is given by ρ. The perturbations due to quasi-solitons are represented by ξ, η and ζ. The fourth and sixth order dispersions are respectively represented by χ and ψ. Finally, the saturable amplifiers are given by σj for j = 1, 2. With these perturbation terms, the perturbed NLSE, given by (10), changes to: −4 2 4 iqt + aqxx + b1 |q| + b2 |q| + b3 |q| q h 2m 2 2 2 = iǫ δ |q| q + αqx + βqxx − γqxxx + λ |q| q + θ |q| q + ρ |qx | q x x 2 q − iξ q 2 qx∗ x − iηqx2 q ∗ − iζq ∗ q 2 xx − iµ |q| x Z x 2 |q| ds (15) − iχqxxxx − iψqxxxxx + (σ1 q + σ2 qx ) −∞
ADIABATIC PARAMETER DYNAMICS
M
3.2
and
te
d
With the perturbed NLSE given by (15), the adiabatic variation of the soliton energy and frequency as indicated by (11) and (12) are given by [5, 8] " ρA4 Γ 12 dE 4δA2m+2 Γ m+1 πβA2 2 B 2 + 8κ2 + B 2 + 12κ2 = ǫ − m dt mB 4B 3B Γ 2 2 4 4 Z ∞ σ1 π A 2σ2 A −1 + − sechs tanh s tan (sinh s) ds , (16) B2 B −∞
Ac ce p
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65
Z ∞ ǫA2 dκ 2 −1 6βκπB − 2µA B + 3σ2 κ sechs tanh s tan (sinh s) ds . = dt 3aE −∞
The velocity change by virtue of (13) takes the form: A2 3γ 2κA2 v = −2aκ − ǫ α + B 2 + κ2 + (3λ + 2θ) + (ξ − η − 2ζ) 8 π π Z 2σ1 A2 ∞ χκ σ2 πA2 B 2 + 8κ2 − + s sechs tan−1 (sinh s) ds + π 2B 2 −∞ 3ψκB − 11 3 + 4κ2 B 4 + 32 1 + 3κ2 B 2 + 128κ4 . 64
(17)
(18)
In order to evaluate the integrals in (11)-(13), the 1-soliton solution given by (2) is utilized.
¯ κ ¯ 0) The dynamical system given by (16) and (17) for µ = ρ = 0 has a stable fixed point given by (A, ¯ ) = (A, ¯ κ ¯ 0) where A¯ and B ¯ are related through the equation [5] or (B, ¯ ) = (B, m+1 Γ 21 2m Γ 2 − mπβB 3 + 4σ1 mπ 2 A2 − 8σ2 mA2 BJ = 0, (19) 16δA B Γ m 2
with
J=
Z
∞
sechs tanh s tan−1 (sinh s) ds.
(20)
−∞
4
Page 4 of 6
4
ip t
This shows that the perturbed soliton moves at the speed of light with a fixed value of the amplitude and width that are given by the solution of (19). This phenomena is known as optical soliton cooling [6].
CONCLUSIONS
an
us
cr
This paper obtained the adiabatic variation of soliton power (energy) and frequency in presence of perturbation terms for anti-cubic nonlinearity. The slow change in soliton velocity is also presented. The corresponding dynamical system led to a stable fixed point and hence the phenomenon of soliton cooling is enlightened for such nonlinearity. These results for anti-cubic nonlinearity are being reported for the first time in this paper. They carry lot of future prospects. The analysis can be extended to other type of nonlinearities that have not been covered. The results of this paper leads to the formulation of quasi-particle theory for supressing the intra-channel soliton-soliton interaction. The outcome of those research is currently awaited, but will be soon disseminated. ACKNOWLEDGEMENTS
te
d
M
The work of the second author (QZ) was supported by the National Science Foundation for Young Scientists of Wuhan Donghu University. The fifth author (SPM) would like to thank the research support provided by the Department of Mathematics and Statistics at Tshwane University of Technology and the support from the South African National Foundation under Grant Number 92052 IRF1202210126. The research work of sixth author (MB) was supported by Qatar National Research Fund (QNRF) under the grant number NPRP 6-021-1-005. The authors also declare that there is no conflict of interest.
Ac ce p
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65
5
Page 5 of 6
References
ip t
[1] N. N. Akhmediev, A. Ankiewicz & R. Grimshaw. “Hamiltonian-versus energy diagrams in soliton theory”. Physical Review E. Volume 59 Issue 5, 6088-6096. (1999). [2] A. Biswas & S. Konar. Introduction to non-Kerr law optical solitons. CRC Press. Boca Raton, FL. (2006).
cr
[3] M. Ekici, M. Mirzazadeh, A. Sonmezoglu, M. Z. Ullah, Q. Zhou, H. Triki, S. P. Moshokoa, & A. Biswas. “Optical solitons with anti-cubic nonlinearity by extended trial equation method”. To appear in Optik.
us
[4] R. Fedele, H. Schamel, V. I. Karpman & P. K. Shukla. “Envelope solitons of nonlinear Schr¨odinger equation with an anti-cubic nonlinearity”. Journal of Physics A. Volume 36, 1169-1173. (2003).
an
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