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Optical solitons in a power law media with fourth order dispersion
Commun Nonlinear Sci Numer Simulat 14 (2009) 1834–1837
Contents lists available at ScienceDirect
Commun Nonlinear Sci Numer Simulat journal homepage: www.elsevier.com/locate/cnsns
Short communication
Optical solitons in a power law media with fourth order dispersion Anjan Biswas a,*, Daniela Milovic b a
Center for Research and Education in Optical Sciences and Applications, Department of Applied Mathematics and Theoretical Physics, Delaware State University, 1200 N. DuPont Highway, Dover, DE 19901-2277, USA Faculty of Electronic Engineering, Department of Telecommunications, University of Nis, Aleksandra Medvedeva 14, 1800 Nis, Serbia
b
a r t i c l e
i n f o
Article history: Received 19 August 2008 Accepted 20 August 2008 Available online 3 September 2008 MSC: 35Q51 35Q53 37K10 78A60
a b s t r a c t In this paper, a closed form optical soliton solution is obtained for the nonlinear Schrödinger’s equation with fourth order dispersion in a power law media. The solitary wave ansatze is used to carry out the integration of this equation. Finally, a numerical simulation is given for the closed form soliton solution. Ó 2008 Elsevier B.V. All rights reserved.
PACS: 02.30.Jr 02.30.Ik Keywords: Optical solitons Power law nonlinearity Fourth order dispersion Integrability
1. Introduction The study of optical solitons has made a remarkable progress in the past few decades. The governing equation in the nonlinear Schrödinger’s equation (NLSE) [1–10]. In general, this equation is studied with various forms of nonlinearity [5] besides the well known Kerr law nonlinearity. In this case, the NLSE is integrable by the classical method of inverse scattering transform (IST) and the equation is well studied. However, in this paper the NLSE is going to be studied with the inclusion of the fourth order dispersion (4OD) term [9]. The NLSE does not give correct prediction for pulse widths smaller than 1 ps. For example, in solid state solitary lasers, where pulses as short as 10 fs are generated, the approximation breaks down. Thus, quasi-monochromaticity is no longer valid and consequently higher order dispersion terms creep in. One needs to consider the higher order dispersion for performance enhancement along trans-oceanic and trans-continental distances. Also, for short pulse widths where the group velocity dispersion changes, within the spectral bandwidth of the signal, can no longer be neglected, one needs to take into account the presence of 4OD [6]. * Corresponding author. Tel.: +1 302 659 0169; fax: +1 302 857 7517. E-mail address: [email protected] (A. Biswas). 1007-5704/$ - see front matter Ó 2008 Elsevier B.V. All rights reserved. doi:10.1016/j.cnsns.2008.08.008
A. Biswas, D. Milovic / Commun Nonlinear Sci Numer Simulat 14 (2009) 1834–1837
1835
2. Mathematical analysis The dimensionless form of the NLSE with 4OD is given by [9]
iqt þ aqxx bqxxxx þ cjqj2m q ¼ 0
ð1Þ
In (1), a, b and c are real numbers. If b = 0, Eq. (1) reduces to the regular NLSE with power law nonlinearity. The coefficient of a represents the group velocity dispersion, while the coefficient of c represents the power law nonlinearity with the exponent m dictating the power law. Also, the coefficient of b is the 4OD term. The solitons are the result of a delicate balance between dispersion and nonlinearity. As mentioned earlier, (1) is not integrable by the classical method of IST. It is still possible to obtain a closed form 1-soliton solution of (1). The method that will be used in this paper is the solitary wave ansatze. It is assumed that the 1-soliton solution to (1) is given by the following phase-amplitude format.
q ¼ Pei/
ð2Þ
where P is the amplitude portion while / is the phase portion of the soliton. It is also assumed that
/ ¼ jx þ xt þ h
ð3Þ
where j is the frequency of the soliton, x is the wave number, while h is the phase constant. Finally, P = P(x, t). On substituting these into (1) yields
oP o/ i/ e iqt ¼ i P ot ot
ð4Þ !
o2 P oP 2ij j2 P ei/ ox2 ox
qxx ¼ qxxx ¼
ð5Þ
! o3 P o2 P 2 oP 3 þ i 3i j 3 j j P ei/ ox3 ox2 ox
qxxxx ¼
! 2 o4 P o3 P 2o P 3 oP 4 4i j 6 j þ 4i j j P ei/ þ ox4 ox3 ox2 ox
ð6Þ ð7Þ
Substituting (4)–(7) into (1) and equating the real and imaginary parts yields
oP oP o3 P 2j a þ 2bj2 þ 4bj 3 ¼ 0 ot ox ox
ð8Þ
and
ðx þ aj2 þ bj4 ÞP cP 2mþ1 ða þ 6bj2 Þ
o2 P o4 P þb 4 ¼0 2 ox ox
ð9Þ
For optical solitons, a judicious choice would be
Pðx; tÞ ¼
A p cosh s
ð10Þ
with
s ¼ Bðx vtÞ
ð11Þ
where 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 derivation of the soliton solution to (1). Thus, from (9) and (10), (8) reduces to
pvAB
tanh s tanh s tanh s 3 tanh s 3 tanh s 3 3 p þ 2ajpAB p þ 4bj pAB p 4bjp AB p þ 4bjpðp þ 1Þðp þ 2ÞAB pþ2 cosh s cosh s cosh s cosh s cosh s
ð12Þ
while (9) reduces to
xA
1 1 1 1 2 2 2 2 2 2 2 p þ ap AB p þ 6bj p AB p j a þ bj p cosh s cosh s cosh s cosh s 1 1 1 1 4 2mþ1 bp AB4 apðp þ 1ÞAB2 6bj2 pðp þ 1ÞAB2 p þ cA pþ2 pþ2 ð2mþ1Þp cosh s cosh s cosh s cosh s 1 1 þ 2bpðp þ 1Þðp2 þ 2p þ 2ÞAB4 bpðp þ 1Þðp þ 2Þðp þ 3ÞAB4 pþ2 pþ4 cosh s cosh s
ð13Þ
1836
A. Biswas, D. Milovic / Commun Nonlinear Sci Numer Simulat 14 (2009) 1834–1837
Fig. 1. Soliton solution of the NLSE for m = 3/2.
Now, from (13) setting the exponents (2m + 1)p and p + 4 equal to one another gives
p¼
2 m
ð14Þ
Also noting that the functions 1/coshps, 1/coshp+2 s and 1/coshp+4s are linearly independent, their respective coefficients in (13) must vanish. Therefore, this yields
" A¼
ðm þ 1Þðm þ 2Þð3m þ 2Þða þ 6bj2 Þ
1 #2m
4bcðm2 þ 2m þ 2Þ2 1 m2 ðq þ 6bj2 Þ 2 B¼ 4bðm2 þ 2m þ 2Þ i 1 h x ¼ 4 m2 j2 ða þ bj2 Þ þ 4m2 B2 ða þ 6bj2 Þ 16bB4 m
ð15Þ
ð16Þ ð17Þ
Finally, applying the same strategy to (12), yields
v¼
m2
2j 2ða þ 6bj2 Þ ðm2 þ 2m þ 2Þða þ 2bj2 Þ þ 2m þ 2
ð18Þ
From (15) and (16), one can conclude that the amplitude and the width of the soliton are related as
"
4bðm þ 1Þðm þ 2Þð3m þ 2ÞB4 A¼ cm4
1 #2m
ð19Þ
Hence, the 1-soliton solution of (1) is given by
qðx; tÞ ¼
A 2
coshm Bðx vtÞ
eiðjxþxtþhÞ
ð20Þ
where the amplitude, width, wave number and the velocity of the soliton are given by (15)–(18), respectively. Fig. 1 shows the profile of a 1-soliton solution with fourth order dispersion for the choice of the parameters a = 1/2, b = c = 1 and m = 3/2. 3. Conclusions In this paper, an exact optical 1-soliton solution to the NLSE with 4OD, for power law nonlinearity, is obtained by the solitary wave ansatze. The governing equation is thus integrable although the Painleve test of integrability will fail. In future, this NLSE will be studied along with its perturbation terms. This will also include the stochastic perturbation terms. The quasi-stationary soliton will be obtained in presence of such perturbation terms.
A. Biswas, D. Milovic / Commun Nonlinear Sci Numer Simulat 14 (2009) 1834–1837
1837
Acknowledgements The research work of the first author (AB) 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 number: W54428-RT-ISP and these supports are genuinely and sincerely appreciated. References [1] Adib B, Heidari A, Tayyari SF. An analytical approach to soliton of the saturable nonlinear Schrödinger equation determination and consideration of stability of solitary solutions of cubic–quintic nonlinear Schrödinger equation (CQNLSE). Commun Nonlinear Sci Numer Simulat 2009;14:2034–45. [2] Akhmediev NN, Ankiewicz A, Grimshaw R. Hamiltonian-versus-energy diagrams in soliton theory. Phys Rev E 1999;59(5):6088–96. [3] Asif N, Shwetanshumala S, Konar S. Photovoltaic spatial soliton pairs in two-photon photorefractive materials. Phys Lett A 2008;372(5):735–40. [4] Busalev VS, Grikurov VE. Simulation of instability of bright solitons for NLS with saturating nonlinearity. Math Comput Simulat 2001;56(6):539–46. [5] Jovanoski Z, Rowland DR. Variational analysis of solitary waves in a homogenous cubic–quintic nonlinear medium. J Mod Opt 2001;48(7):1179–93. [6] Kohl R, Biswas A, Milovic D, Zerrad E. Optical soliton perturbation in a non-Kerr law media. Opt Laser Technol 2008;40(4):647–62. [7] Lin C, Hong X-R, Dou F-Q. Analytical studies of the steady solution for optical soliton equation with higher order dispersion and nonlinear effects. Commun Nonlinear Sci Numer Simulat 2008;13(3):567–74. [8] Shwetanshumala S, Konar S, Biswas A. Spatial optical solitons in inhomogeneous elliptic core saturating nonlinear fiber. Optik 2008;119(9):403–8. [9] Wazwaz AM. Exact solutions for the fourth order nonlinear Schrödinger equations with cubic and power law nonlinearities. Math Comput Modell 2006;43(7–8):802–8. [10] Zhidkov PE. Korteweg-de Vries nonlinear Schrödingers equations qualitative theory. New York (NY): Springer-Verlag; 2001.
Contents lists available at ScienceDirect
Commun Nonlinear Sci Numer Simulat journal homepage: www.elsevier.com/locate/cnsns
Short communication
Optical solitons in a power law media with fourth order dispersion Anjan Biswas a,*, Daniela Milovic b a
Center for Research and Education in Optical Sciences and Applications, Department of Applied Mathematics and Theoretical Physics, Delaware State University, 1200 N. DuPont Highway, Dover, DE 19901-2277, USA Faculty of Electronic Engineering, Department of Telecommunications, University of Nis, Aleksandra Medvedeva 14, 1800 Nis, Serbia
b
a r t i c l e
i n f o
Article history: Received 19 August 2008 Accepted 20 August 2008 Available online 3 September 2008 MSC: 35Q51 35Q53 37K10 78A60
a b s t r a c t In this paper, a closed form optical soliton solution is obtained for the nonlinear Schrödinger’s equation with fourth order dispersion in a power law media. The solitary wave ansatze is used to carry out the integration of this equation. Finally, a numerical simulation is given for the closed form soliton solution. Ó 2008 Elsevier B.V. All rights reserved.
PACS: 02.30.Jr 02.30.Ik Keywords: Optical solitons Power law nonlinearity Fourth order dispersion Integrability
1. Introduction The study of optical solitons has made a remarkable progress in the past few decades. The governing equation in the nonlinear Schrödinger’s equation (NLSE) [1–10]. In general, this equation is studied with various forms of nonlinearity [5] besides the well known Kerr law nonlinearity. In this case, the NLSE is integrable by the classical method of inverse scattering transform (IST) and the equation is well studied. However, in this paper the NLSE is going to be studied with the inclusion of the fourth order dispersion (4OD) term [9]. The NLSE does not give correct prediction for pulse widths smaller than 1 ps. For example, in solid state solitary lasers, where pulses as short as 10 fs are generated, the approximation breaks down. Thus, quasi-monochromaticity is no longer valid and consequently higher order dispersion terms creep in. One needs to consider the higher order dispersion for performance enhancement along trans-oceanic and trans-continental distances. Also, for short pulse widths where the group velocity dispersion changes, within the spectral bandwidth of the signal, can no longer be neglected, one needs to take into account the presence of 4OD [6]. * Corresponding author. Tel.: +1 302 659 0169; fax: +1 302 857 7517. E-mail address: [email protected] (A. Biswas). 1007-5704/$ - see front matter Ó 2008 Elsevier B.V. All rights reserved. doi:10.1016/j.cnsns.2008.08.008
A. Biswas, D. Milovic / Commun Nonlinear Sci Numer Simulat 14 (2009) 1834–1837
1835
2. Mathematical analysis The dimensionless form of the NLSE with 4OD is given by [9]
iqt þ aqxx bqxxxx þ cjqj2m q ¼ 0
ð1Þ
In (1), a, b and c are real numbers. If b = 0, Eq. (1) reduces to the regular NLSE with power law nonlinearity. The coefficient of a represents the group velocity dispersion, while the coefficient of c represents the power law nonlinearity with the exponent m dictating the power law. Also, the coefficient of b is the 4OD term. The solitons are the result of a delicate balance between dispersion and nonlinearity. As mentioned earlier, (1) is not integrable by the classical method of IST. It is still possible to obtain a closed form 1-soliton solution of (1). The method that will be used in this paper is the solitary wave ansatze. It is assumed that the 1-soliton solution to (1) is given by the following phase-amplitude format.
q ¼ Pei/
ð2Þ
where P is the amplitude portion while / is the phase portion of the soliton. It is also assumed that
/ ¼ jx þ xt þ h
ð3Þ
where j is the frequency of the soliton, x is the wave number, while h is the phase constant. Finally, P = P(x, t). On substituting these into (1) yields
oP o/ i/ e iqt ¼ i P ot ot
ð4Þ !
o2 P oP 2ij j2 P ei/ ox2 ox
qxx ¼ qxxx ¼
ð5Þ
! o3 P o2 P 2 oP 3 þ i 3i j 3 j j P ei/ ox3 ox2 ox
qxxxx ¼
! 2 o4 P o3 P 2o P 3 oP 4 4i j 6 j þ 4i j j P ei/ þ ox4 ox3 ox2 ox
ð6Þ ð7Þ
Substituting (4)–(7) into (1) and equating the real and imaginary parts yields
oP oP o3 P 2j a þ 2bj2 þ 4bj 3 ¼ 0 ot ox ox
ð8Þ
and
ðx þ aj2 þ bj4 ÞP cP 2mþ1 ða þ 6bj2 Þ
o2 P o4 P þb 4 ¼0 2 ox ox
ð9Þ
For optical solitons, a judicious choice would be
Pðx; tÞ ¼
A p cosh s
ð10Þ
with
s ¼ Bðx vtÞ
ð11Þ
where 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 derivation of the soliton solution to (1). Thus, from (9) and (10), (8) reduces to
pvAB
tanh s tanh s tanh s 3 tanh s 3 tanh s 3 3 p þ 2ajpAB p þ 4bj pAB p 4bjp AB p þ 4bjpðp þ 1Þðp þ 2ÞAB pþ2 cosh s cosh s cosh s cosh s cosh s
ð12Þ
while (9) reduces to
xA
1 1 1 1 2 2 2 2 2 2 2 p þ ap AB p þ 6bj p AB p j a þ bj p cosh s cosh s cosh s cosh s 1 1 1 1 4 2mþ1 bp AB4 apðp þ 1ÞAB2 6bj2 pðp þ 1ÞAB2 p þ cA pþ2 pþ2 ð2mþ1Þp cosh s cosh s cosh s cosh s 1 1 þ 2bpðp þ 1Þðp2 þ 2p þ 2ÞAB4 bpðp þ 1Þðp þ 2Þðp þ 3ÞAB4 pþ2 pþ4 cosh s cosh s
ð13Þ
1836
A. Biswas, D. Milovic / Commun Nonlinear Sci Numer Simulat 14 (2009) 1834–1837
Fig. 1. Soliton solution of the NLSE for m = 3/2.
Now, from (13) setting the exponents (2m + 1)p and p + 4 equal to one another gives
p¼
2 m
ð14Þ
Also noting that the functions 1/coshps, 1/coshp+2 s and 1/coshp+4s are linearly independent, their respective coefficients in (13) must vanish. Therefore, this yields
" A¼
ðm þ 1Þðm þ 2Þð3m þ 2Þða þ 6bj2 Þ
1 #2m
4bcðm2 þ 2m þ 2Þ2 1 m2 ðq þ 6bj2 Þ 2 B¼ 4bðm2 þ 2m þ 2Þ i 1 h x ¼ 4 m2 j2 ða þ bj2 Þ þ 4m2 B2 ða þ 6bj2 Þ 16bB4 m
ð15Þ
ð16Þ ð17Þ
Finally, applying the same strategy to (12), yields
v¼
m2
2j 2ða þ 6bj2 Þ ðm2 þ 2m þ 2Þða þ 2bj2 Þ þ 2m þ 2
ð18Þ
From (15) and (16), one can conclude that the amplitude and the width of the soliton are related as
"
4bðm þ 1Þðm þ 2Þð3m þ 2ÞB4 A¼ cm4
1 #2m
ð19Þ
Hence, the 1-soliton solution of (1) is given by
qðx; tÞ ¼
A 2
coshm Bðx vtÞ
eiðjxþxtþhÞ
ð20Þ
where the amplitude, width, wave number and the velocity of the soliton are given by (15)–(18), respectively. Fig. 1 shows the profile of a 1-soliton solution with fourth order dispersion for the choice of the parameters a = 1/2, b = c = 1 and m = 3/2. 3. Conclusions In this paper, an exact optical 1-soliton solution to the NLSE with 4OD, for power law nonlinearity, is obtained by the solitary wave ansatze. The governing equation is thus integrable although the Painleve test of integrability will fail. In future, this NLSE will be studied along with its perturbation terms. This will also include the stochastic perturbation terms. The quasi-stationary soliton will be obtained in presence of such perturbation terms.
A. Biswas, D. Milovic / Commun Nonlinear Sci Numer Simulat 14 (2009) 1834–1837
1837
Acknowledgements The research work of the first author (AB) 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 number: W54428-RT-ISP and these supports are genuinely and sincerely appreciated. References [1] Adib B, Heidari A, Tayyari SF. An analytical approach to soliton of the saturable nonlinear Schrödinger equation determination and consideration of stability of solitary solutions of cubic–quintic nonlinear Schrödinger equation (CQNLSE). Commun Nonlinear Sci Numer Simulat 2009;14:2034–45. [2] Akhmediev NN, Ankiewicz A, Grimshaw R. Hamiltonian-versus-energy diagrams in soliton theory. Phys Rev E 1999;59(5):6088–96. [3] Asif N, Shwetanshumala S, Konar S. Photovoltaic spatial soliton pairs in two-photon photorefractive materials. Phys Lett A 2008;372(5):735–40. [4] Busalev VS, Grikurov VE. Simulation of instability of bright solitons for NLS with saturating nonlinearity. Math Comput Simulat 2001;56(6):539–46. [5] Jovanoski Z, Rowland DR. Variational analysis of solitary waves in a homogenous cubic–quintic nonlinear medium. J Mod Opt 2001;48(7):1179–93. [6] Kohl R, Biswas A, Milovic D, Zerrad E. Optical soliton perturbation in a non-Kerr law media. Opt Laser Technol 2008;40(4):647–62. [7] Lin C, Hong X-R, Dou F-Q. Analytical studies of the steady solution for optical soliton equation with higher order dispersion and nonlinear effects. Commun Nonlinear Sci Numer Simulat 2008;13(3):567–74. [8] Shwetanshumala S, Konar S, Biswas A. Spatial optical solitons in inhomogeneous elliptic core saturating nonlinear fiber. Optik 2008;119(9):403–8. [9] Wazwaz AM. Exact solutions for the fourth order nonlinear Schrödinger equations with cubic and power law nonlinearities. Math Comput Modell 2006;43(7–8):802–8. [10] Zhidkov PE. Korteweg-de Vries nonlinear Schrödingers equations qualitative theory. New York (NY): Springer-Verlag; 2001.