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1-soliton solution of the generalized Radhakrishnan, Kundu, Lakshmanan equation
Physics Letters A 373 (2009) 2546–2548
Contents lists available at ScienceDirect
Physics Letters A www.elsevier.com/locate/pla
1-soliton solution of the generalized Radhakrishnan, Kundu, Lakshmanan equation Anjan Biswas Center for Research and Education in Optical Sciences and Applications, Department of Applied Mathematics and Theoretical Physics, Delaware State University, Dover, DE 19901-2277, USA
a r t i c l e
i n f o
Article history: Received 21 March 2009 Received in revised form 30 April 2009 Accepted 8 May 2009 Available online 15 May 2009 Communicated by R. Wu
a b s t r a c t This Letter carries out the integration of the generalized Radhakrishnan, Kundu, Lakshmanan equation to obtain the 1-soliton solution. The solitary wave ansatz is used to carry out the integration, to obtain an exact solution of this equation. © 2009 Elsevier B.V. All rights reserved.
PACS: 02.30.Jr 02.30.Ik MSC: 35Q51 35Q53 37K10 78A60 Keywords: Optical solitons Integrability Exact solution
1. Introduction
2. Governing equation
The propagation of solitons through optical fibers has been a major research area in the past few decades [1–15]. This area of research made remarkable progress during this time period. Today, although optical solitons are a reality in many parts of the world, the research in this area has not slowed down. There are many new results that are being constantly reported, in various journals, in this area. One of the major issue about the theory of optical solitons is the integrability of governing equation. Although there are several numerical schemes that can solve these equation numerically, so that it is possible to visualize the solution, a closed form exact solution is still necessary for analytical study of the properties of the solitons. There are many modern methods of integrability of these equations that describe the dynamics of solitons [10]. In this Letter, the soliton ansatz is going to be implemented to carry out the integration of the governing equation. It needs to be noted that this method of soliton ansatz is very similar to the exponential function method [2–4].
The governing equation, in dimensionless form, for the propagation of solitons through an optical fiber is given by [4,15]
E-mail address: [email protected]. 0375-9601/$ – see front matter © 2009 Elsevier B.V. All rights reserved. doi:10.1016/j.physleta.2009.05.010
iqt + aq xx + b|q|2m q = i λ |q|2m q
x
− i γ q xxx
(1)
where q(x, 0) =
A cosh Bx
,
−∞ < x < +∞.
(2)
In (1), a and b are real numbers. Here, a represents the coefficient of group velocity dispersion term and b represents the coefficient of nonlinearity. This is the case of power law nonlinearity which is a general form of the usually studied Kerr law nonlinearity. The parameter m in (1) dictates the power law nonlinearity. It needs to be noted that for stability of the soliton solutions of (1), it is necessary to have 0 < m < 2 [2,6,7]. Physically, power law nonlinearity arises in various materials, including semiconductors. Moreover, this law of nonlinearity arises in nonlinear plasmas that solves the problem of small K -condensation in weak turbulence theory [1,4]. Solitons are the result of a delicate balance between dispersion and nonlinearity [4,7,8].
A. Biswas / Physics Letters A 373 (2009) 2546–2548
On the right-hand side of (1), the coefficient of λ represents the self-steepening term for short pulses [1,4] (typically 100 fs). The coefficient of third order dispersion term is given by γ . It is known that (1) does not give correct prediction for pulse widths smaller than 1 picosecond. For example, in solid state solitary lasers, where pulses as short as 10 femtoseconds are generated, the approximation breaks down. Thus, quasi-monochromaticity is no longer valid and so higher order dispersion terms come in. If the group velocity dispersion is close to zero, one needs to consider the third order dispersion for performance enhancement along trans-oceanic and trans-continental distances. Also, for short pulse widths where group velocity dispersion changes, within the spectral bandwidth of the signal, cannot be neglected, one needs to take into account the presence of higher order dispersion terms [1,4]. Eq. (1) is commonly known as the generalized form Radhakrishnan, Kundu, Lakshmanan (RKL) equation as coined in 2008 [15]. The special case where m = 1 in (1) is known as the RKL equation [4]. It is known that this equation supports solitons that are studied in the context of nonlinear fiber optics. Now, Eq. (1) will be integrated to obtain the exact 1-soliton solution for any arbitrary m.
2547
derivation of the soliton solution to (1). Thus, from (10) and (11), Eqs. (8) and (9) respectively reduce to
ω + aκ 2 + γ κ 3 A + (a + 3γ κ ) p 2 A B 2 + (a + 3γ κ ) p ( p + 1) A B 2 + (λκ − b) A 2m+1
1 cosh p τ
1 cosh p +2 τ
1 cosh(2m+1) p τ
= 0,
(12)
vp A B + 2aκ p A B − γ p 3 A B 3 + 3γ κ 2 p A B
+ γ p ( p + 1)( p + 2) A B 3 + λ(2m + 1) p A 2m+1 B
tanh τ cosh p τ
tanh τ cosh p +2 τ tanh τ
cosh(2m+1) p τ
= 0.
(13)
Now, from (12) or (13), equating the exponents (2m + 1) p and ( p + 2) gives
(2m + 1) p = p + 2
(14)
that gives 1
3. Mathematical analysis
p=
In order to solve (1), it is first necessary to write the solution in the phase-amplitude format as
From (13), setting the coefficients of 1/ cosh p + j τ , for j = 1, 2 to zero, as these are linearly independent functions leads to
q = P eiφ
v =−
(3)
m
1 2 2m aκ − γ B 2 + 3m2 γ κ 2
m2
where P is the amplitude portion while φ is the phase portion of the soliton. It is also assumed that
and
φ = −κ x + ωt + θ
B= −
(4)
where κ is the frequency of the soliton, ω is the wave number, while θ is the phase constant. Finally, P = P (x, t ), which represents the pulse shape. On substituting these into (1) yields
iqt = i
∂P ∂φ i φ −P e , ∂t ∂t
(5)
∂2 P ∂P − 2i κ − κ 2 P eiφ , ∂x ∂ x2 3 ∂2 P ∂ P 2∂P 3 q xxx = − 3i κ − 3 κ + i κ P eiφ . ∂x ∂ x3 ∂ x2 q xx =
(15)
.
λm2 Am γ (m + 1)
12 (17)
.
Similarly, from (12), it is possible to obtain
ω=−
(a + 3γ κ ) B 2 − m2 κ 2 (a + γ κ )
1
m2
A=
(6)
λ(a + 3γ κ ) γ (λκ − b)
m1 (19)
so that from (17) and (19) it is possible to recover (7)
B= −
λ2m2 (a + 3γ κ ) 2 γ (m + 1)(λκ − b)
12 .
The wave number of the soliton, by virtue of (20) reduces to
ω=
3
ω + aκ + γ κ P − (b − λκ ) P
2m+1
∂2 P − (a + 3γ κ ) 2 = 0 ∂x
(8)
and
∂P ∂P ∂P ∂3 P − 2aκ + 3γ κ 2 − λ(2m + 1) P 2m − γ 3 = 0. ∂t ∂x ∂x ∂x
(9)
A p
cosh B (x − vt )
=
A cosh
p
τ
(10)
where
τ = B (x − vt )
(11)
and A is the amplitude, B is the inverse width of the soliton and v is the soliton velocity, while the exponent p is unknown at this stage. This unknown exponent will be determined in the course of
(20)
1
(m + 1)γ 2 (λκ − b)
× λ2 (a + 3γ κ )2 + (m + 1)κ 2 γ 2 (λκ − b)(a + γ κ )
(21)
which is free from the soliton amplitude and width. It needs to be noted from (17) and (20) respectively that the solitons will exist for the RKL equation provided
λγ < 0
For optical solitons, a judicious choice would be [1,2,6–11] P (x, t ) =
(18)
and
Substituting (3)–(7) into (1) and equating the real and imaginary parts respectively yields 2
(16)
(22)
and
(a + 3γ κ )(λκ − b) < 0
(23)
which are the corresponding domain restrictions for the existence of the soliton. Thus, the 1-soliton solution of the generalized RKL equation is given by q=
A 1
cosh m
e i (−κ x+ωt +θ)
τ
(24)
2548
A. Biswas / Physics Letters A 373 (2009) 2546–2548
where the amplitude A, the inverse width of the soliton B, the velocity v of the soliton and the wave number ω of the soliton are respectively given by (19), (20), (16) and (21). The relation (17) gives a direct connection between the amplitude and width.
ues. These results in (27)–(31) agree with the known results that are studied in this context earlier by various other analytical techniques for example, Lie transform, Lie group analysis or He’s variational principle [1,5,7].
3.1. Kerr law nonlinearity
4. Conclusions
The special case where m = 1 is called Kerr law nonlinearity. The Kerr law of nonlinearity originates from the fact that a light wave in an optical fiber faces nonlinear responses from nonharmonic motion of electrons bound in molecules, caused by an external electric field. Even though the nonlinear responses are extremely weak, their effects appear in various ways over long distance of propagation that is measured in terms of light wavelength. The origin of nonlinear response is related to the non-harmonic motion of bound electrons under the influence of an applied field. As a result, the induced polarization is not linear in the electric field, but involves higher order terms in electric field amplitude. Thus, the RKL equation with Kerr law nonlinearity reduces to
In this Letter, an exact 1-soliton solution to the generalized RKL equation is obtained by the solitary wave ansatz method. This soliton solution can be used to carry out further studies of this equation. Some of the aspects that will be touched upon are the four-wave mixing, collision-induced timing jitter, stochastic perturbation, time-dependent dispersion and nonlinearity and many more. Furthermore, the quasi-particle theory of the RKL equation will be developed. The results of those research will be reported in future.
iqt + aq xx + b|q|2 q = i λ |q|2 q
The research work was fully supported by NSF-CREST Grant No. HRD-0630388 and Army Research Office (ARO) along with the Air Force Office of Scientific Research (AFOSR) under the award No. W54428-RT-ISP and these supports are genuinely and sincerely appreciated.
x
− i γ q xxx .
(25)
Now, from (15), p = m = 1.
(26)
Hence, the soliton parameters in this case gives
v = − 2aκ − γ B 2 + 3γ κ 2 , λA B= − , 2γ
ω = − B 2 (a + 3γ κ ) − κ 2 (a + γ κ ) , λ(a + 3γ κ ) A= , γ (λκ − b) 2 1 λ (a + 3γ κ ) 2 . B= − 2γ 2 (λκ − b)
References (27) (28) (29) (30)
(31)
Thus from (31), the wave number simplifies to
ω=
1 2γ
2 (λ
κ − b)
λ2 (a + 3γ κ )2 + 2κ 2 γ 2 (λκ − b)(a + γ κ ) .
Acknowledgements
(32)
The special case of power law nonlinearity, that is the Kerr law nonlinearity obtained by setting m = 1 has these parameter val-
[1] A. Biswas, Opt. Quantum Electron. 35 (10) (2003) 979. [2] D.D. Ganji, M. Abdollahzadeh, Appl. Math. Comput. 206 (1) (2008) 438. [3] Z.Z. Ganji, D.D. Gangi, A. Asgari, Finding general and explicit solutions of higher nonlinear equations by the Exp-Function method, Comput. Math. Appl., in press. [4] D.D. Ganji, A. Asgari, Z.Z. Ganji, Acta Appl. Math. 104 (2) (2008) 201. [5] C.M. Khalique, A. Biswas, J. Electromagnetic Waves Appl. 23 (7) (2009) 963. [6] R. Kohl, A. Biswas, D. Milovic, E. Zerrad, Opt. Laser Technol. 40 (4) (2008) 647. [7] R. Kohl, D. Milovic, E. Zerrad, A. Biswas, J. Infrared Millimeter Terrahertz Waves 30 (5) (2009) 526. [8] S. Konar, M. Mishra, S. Jana, Phys. Lett. A 362 (5–6) (2007) 505. [9] S. Konar, S. Jana, W.-P. Hong, Phys. Scr. 76 (5) (2007) 470. [10] W.-P. Hong, Opt. Commun. 194 (1–3) (2001) 217. [11] W.-P. Hong, S. Shwetanshumala, S. Konar, Opt. Commun. 281 (23) (2008) 5864. [12] N.A. Kudryashov, Commun. Nonlinear Sci. Numer. Simul. 14 (9–10) (2009) 3507. [13] N.A. Kudryashov, Commun. Nonlinear Sci. Numer. Simul. 14 (5) (2009) 1881. [14] M. Wang, X. Li, J. Zhang, Phys. Lett. A 363 (1–2) (2007) 96. [15] J.-L. Zhang, M.-L. Wang, Chaos Solitons Fractals 37 (1) (2008) 215.
Contents lists available at ScienceDirect
Physics Letters A www.elsevier.com/locate/pla
1-soliton solution of the generalized Radhakrishnan, Kundu, Lakshmanan equation Anjan Biswas Center for Research and Education in Optical Sciences and Applications, Department of Applied Mathematics and Theoretical Physics, Delaware State University, Dover, DE 19901-2277, USA
a r t i c l e
i n f o
Article history: Received 21 March 2009 Received in revised form 30 April 2009 Accepted 8 May 2009 Available online 15 May 2009 Communicated by R. Wu
a b s t r a c t This Letter carries out the integration of the generalized Radhakrishnan, Kundu, Lakshmanan equation to obtain the 1-soliton solution. The solitary wave ansatz is used to carry out the integration, to obtain an exact solution of this equation. © 2009 Elsevier B.V. All rights reserved.
PACS: 02.30.Jr 02.30.Ik MSC: 35Q51 35Q53 37K10 78A60 Keywords: Optical solitons Integrability Exact solution
1. Introduction
2. Governing equation
The propagation of solitons through optical fibers has been a major research area in the past few decades [1–15]. This area of research made remarkable progress during this time period. Today, although optical solitons are a reality in many parts of the world, the research in this area has not slowed down. There are many new results that are being constantly reported, in various journals, in this area. One of the major issue about the theory of optical solitons is the integrability of governing equation. Although there are several numerical schemes that can solve these equation numerically, so that it is possible to visualize the solution, a closed form exact solution is still necessary for analytical study of the properties of the solitons. There are many modern methods of integrability of these equations that describe the dynamics of solitons [10]. In this Letter, the soliton ansatz is going to be implemented to carry out the integration of the governing equation. It needs to be noted that this method of soliton ansatz is very similar to the exponential function method [2–4].
The governing equation, in dimensionless form, for the propagation of solitons through an optical fiber is given by [4,15]
E-mail address: [email protected]. 0375-9601/$ – see front matter © 2009 Elsevier B.V. All rights reserved. doi:10.1016/j.physleta.2009.05.010
iqt + aq xx + b|q|2m q = i λ |q|2m q
x
− i γ q xxx
(1)
where q(x, 0) =
A cosh Bx
,
−∞ < x < +∞.
(2)
In (1), a and b are real numbers. Here, a represents the coefficient of group velocity dispersion term and b represents the coefficient of nonlinearity. This is the case of power law nonlinearity which is a general form of the usually studied Kerr law nonlinearity. The parameter m in (1) dictates the power law nonlinearity. It needs to be noted that for stability of the soliton solutions of (1), it is necessary to have 0 < m < 2 [2,6,7]. Physically, power law nonlinearity arises in various materials, including semiconductors. Moreover, this law of nonlinearity arises in nonlinear plasmas that solves the problem of small K -condensation in weak turbulence theory [1,4]. Solitons are the result of a delicate balance between dispersion and nonlinearity [4,7,8].
A. Biswas / Physics Letters A 373 (2009) 2546–2548
On the right-hand side of (1), the coefficient of λ represents the self-steepening term for short pulses [1,4] (typically 100 fs). The coefficient of third order dispersion term is given by γ . It is known that (1) does not give correct prediction for pulse widths smaller than 1 picosecond. For example, in solid state solitary lasers, where pulses as short as 10 femtoseconds are generated, the approximation breaks down. Thus, quasi-monochromaticity is no longer valid and so higher order dispersion terms come in. If the group velocity dispersion is close to zero, one needs to consider the third order dispersion for performance enhancement along trans-oceanic and trans-continental distances. Also, for short pulse widths where group velocity dispersion changes, within the spectral bandwidth of the signal, cannot be neglected, one needs to take into account the presence of higher order dispersion terms [1,4]. Eq. (1) is commonly known as the generalized form Radhakrishnan, Kundu, Lakshmanan (RKL) equation as coined in 2008 [15]. The special case where m = 1 in (1) is known as the RKL equation [4]. It is known that this equation supports solitons that are studied in the context of nonlinear fiber optics. Now, Eq. (1) will be integrated to obtain the exact 1-soliton solution for any arbitrary m.
2547
derivation of the soliton solution to (1). Thus, from (10) and (11), Eqs. (8) and (9) respectively reduce to
ω + aκ 2 + γ κ 3 A + (a + 3γ κ ) p 2 A B 2 + (a + 3γ κ ) p ( p + 1) A B 2 + (λκ − b) A 2m+1
1 cosh p τ
1 cosh p +2 τ
1 cosh(2m+1) p τ
= 0,
(12)
vp A B + 2aκ p A B − γ p 3 A B 3 + 3γ κ 2 p A B
+ γ p ( p + 1)( p + 2) A B 3 + λ(2m + 1) p A 2m+1 B
tanh τ cosh p τ
tanh τ cosh p +2 τ tanh τ
cosh(2m+1) p τ
= 0.
(13)
Now, from (12) or (13), equating the exponents (2m + 1) p and ( p + 2) gives
(2m + 1) p = p + 2
(14)
that gives 1
3. Mathematical analysis
p=
In order to solve (1), it is first necessary to write the solution in the phase-amplitude format as
From (13), setting the coefficients of 1/ cosh p + j τ , for j = 1, 2 to zero, as these are linearly independent functions leads to
q = P eiφ
v =−
(3)
m
1 2 2m aκ − γ B 2 + 3m2 γ κ 2
m2
where P is the amplitude portion while φ is the phase portion of the soliton. It is also assumed that
and
φ = −κ x + ωt + θ
B= −
(4)
where κ is the frequency of the soliton, ω is the wave number, while θ is the phase constant. Finally, P = P (x, t ), which represents the pulse shape. On substituting these into (1) yields
iqt = i
∂P ∂φ i φ −P e , ∂t ∂t
(5)
∂2 P ∂P − 2i κ − κ 2 P eiφ , ∂x ∂ x2 3 ∂2 P ∂ P 2∂P 3 q xxx = − 3i κ − 3 κ + i κ P eiφ . ∂x ∂ x3 ∂ x2 q xx =
(15)
.
λm2 Am γ (m + 1)
12 (17)
.
Similarly, from (12), it is possible to obtain
ω=−
(a + 3γ κ ) B 2 − m2 κ 2 (a + γ κ )
1
m2
A=
(6)
λ(a + 3γ κ ) γ (λκ − b)
m1 (19)
so that from (17) and (19) it is possible to recover (7)
B= −
λ2m2 (a + 3γ κ ) 2 γ (m + 1)(λκ − b)
12 .
The wave number of the soliton, by virtue of (20) reduces to
ω=
3
ω + aκ + γ κ P − (b − λκ ) P
2m+1
∂2 P − (a + 3γ κ ) 2 = 0 ∂x
(8)
and
∂P ∂P ∂P ∂3 P − 2aκ + 3γ κ 2 − λ(2m + 1) P 2m − γ 3 = 0. ∂t ∂x ∂x ∂x
(9)
A p
cosh B (x − vt )
=
A cosh
p
τ
(10)
where
τ = B (x − vt )
(11)
and A is the amplitude, B is the inverse width of the soliton and v is the soliton velocity, while the exponent p is unknown at this stage. This unknown exponent will be determined in the course of
(20)
1
(m + 1)γ 2 (λκ − b)
× λ2 (a + 3γ κ )2 + (m + 1)κ 2 γ 2 (λκ − b)(a + γ κ )
(21)
which is free from the soliton amplitude and width. It needs to be noted from (17) and (20) respectively that the solitons will exist for the RKL equation provided
λγ < 0
For optical solitons, a judicious choice would be [1,2,6–11] P (x, t ) =
(18)
and
Substituting (3)–(7) into (1) and equating the real and imaginary parts respectively yields 2
(16)
(22)
and
(a + 3γ κ )(λκ − b) < 0
(23)
which are the corresponding domain restrictions for the existence of the soliton. Thus, the 1-soliton solution of the generalized RKL equation is given by q=
A 1
cosh m
e i (−κ x+ωt +θ)
τ
(24)
2548
A. Biswas / Physics Letters A 373 (2009) 2546–2548
where the amplitude A, the inverse width of the soliton B, the velocity v of the soliton and the wave number ω of the soliton are respectively given by (19), (20), (16) and (21). The relation (17) gives a direct connection between the amplitude and width.
ues. These results in (27)–(31) agree with the known results that are studied in this context earlier by various other analytical techniques for example, Lie transform, Lie group analysis or He’s variational principle [1,5,7].
3.1. Kerr law nonlinearity
4. Conclusions
The special case where m = 1 is called Kerr law nonlinearity. The Kerr law of nonlinearity originates from the fact that a light wave in an optical fiber faces nonlinear responses from nonharmonic motion of electrons bound in molecules, caused by an external electric field. Even though the nonlinear responses are extremely weak, their effects appear in various ways over long distance of propagation that is measured in terms of light wavelength. The origin of nonlinear response is related to the non-harmonic motion of bound electrons under the influence of an applied field. As a result, the induced polarization is not linear in the electric field, but involves higher order terms in electric field amplitude. Thus, the RKL equation with Kerr law nonlinearity reduces to
In this Letter, an exact 1-soliton solution to the generalized RKL equation is obtained by the solitary wave ansatz method. This soliton solution can be used to carry out further studies of this equation. Some of the aspects that will be touched upon are the four-wave mixing, collision-induced timing jitter, stochastic perturbation, time-dependent dispersion and nonlinearity and many more. Furthermore, the quasi-particle theory of the RKL equation will be developed. The results of those research will be reported in future.
iqt + aq xx + b|q|2 q = i λ |q|2 q
The research work was fully supported by NSF-CREST Grant No. HRD-0630388 and Army Research Office (ARO) along with the Air Force Office of Scientific Research (AFOSR) under the award No. W54428-RT-ISP and these supports are genuinely and sincerely appreciated.
x
− i γ q xxx .
(25)
Now, from (15), p = m = 1.
(26)
Hence, the soliton parameters in this case gives
v = − 2aκ − γ B 2 + 3γ κ 2 , λA B= − , 2γ
ω = − B 2 (a + 3γ κ ) − κ 2 (a + γ κ ) , λ(a + 3γ κ ) A= , γ (λκ − b) 2 1 λ (a + 3γ κ ) 2 . B= − 2γ 2 (λκ − b)
References (27) (28) (29) (30)
(31)
Thus from (31), the wave number simplifies to
ω=
1 2γ
2 (λ
κ − b)
λ2 (a + 3γ κ )2 + 2κ 2 γ 2 (λκ − b)(a + γ κ ) .
Acknowledgements
(32)
The special case of power law nonlinearity, that is the Kerr law nonlinearity obtained by setting m = 1 has these parameter val-
[1] A. Biswas, Opt. Quantum Electron. 35 (10) (2003) 979. [2] D.D. Ganji, M. Abdollahzadeh, Appl. Math. Comput. 206 (1) (2008) 438. [3] Z.Z. Ganji, D.D. Gangi, A. Asgari, Finding general and explicit solutions of higher nonlinear equations by the Exp-Function method, Comput. Math. Appl., in press. [4] D.D. Ganji, A. Asgari, Z.Z. Ganji, Acta Appl. Math. 104 (2) (2008) 201. [5] C.M. Khalique, A. Biswas, J. Electromagnetic Waves Appl. 23 (7) (2009) 963. [6] R. Kohl, A. Biswas, D. Milovic, E. Zerrad, Opt. Laser Technol. 40 (4) (2008) 647. [7] R. Kohl, D. Milovic, E. Zerrad, A. Biswas, J. Infrared Millimeter Terrahertz Waves 30 (5) (2009) 526. [8] S. Konar, M. Mishra, S. Jana, Phys. Lett. A 362 (5–6) (2007) 505. [9] S. Konar, S. Jana, W.-P. Hong, Phys. Scr. 76 (5) (2007) 470. [10] W.-P. Hong, Opt. Commun. 194 (1–3) (2001) 217. [11] W.-P. Hong, S. Shwetanshumala, S. Konar, Opt. Commun. 281 (23) (2008) 5864. [12] N.A. Kudryashov, Commun. Nonlinear Sci. Numer. Simul. 14 (9–10) (2009) 3507. [13] N.A. Kudryashov, Commun. Nonlinear Sci. Numer. Simul. 14 (5) (2009) 1881. [14] M. Wang, X. Li, J. Zhang, Phys. Lett. A 363 (1–2) (2007) 96. [15] J.-L. Zhang, M.-L. Wang, Chaos Solitons Fractals 37 (1) (2008) 215.