Lax Pair, Conservation Laws, Solitons, and Rogue Waves for a Generalised Nonlinear Schrödinger–Maxwell–Bloch System under the Nonlinear Tunneling Effect for an Inhomogeneous Erbium-Doped Silica Fibre
-
Zhe Gao
,
Chuan-Qi Su
Abstract
Under investigation in this article is a generalised nonlinear Schrödinger-Maxwell-Bloch system for the picosecond optical pulse propagation in an inhomogeneous erbium-doped silica optical fibre. Lax pair, conservation laws, Darboux transformation, and generalised Darboux transformation for the system are constructed; with the one- and two-soliton solutions, the first- and second-order rogue waves given. Soliton propagation is discussed. Nonlinear tunneling effect on the solitons and rogue waves are investigated. We find that (i) the detuning of the atomic transition frequency from the optical pulse frequency affects the velocity of the pulse when the detuning is small, (ii) nonlinear tunneling effect does not affect the energy redistribution of the soliton interaction, (iii) dispersion barrier/well has an effect on the soliton velocity, whereas nonlinear well/barrier does not, (iv) nonlinear well/barrier could amplify/compress the solitons or rogue waves in a smoother manner than the dispersion barrier/well, and (v) dispersion barrier could “attract” the nearby rogue waves, whereas the dispersion well has a repulsive effect on them.
Acknowledgments
We express our sincere thanks to the editors, referees, and all the members of our discussion group for their valuable comments. This work has been supported by the National Natural Science Foundation of China under Grant No. 11272023 and by the Open Fund of State Key Laboratory of Information Photonics and Optical Communications (Beijing University of Posts and Telecommunications) under Grant No. IPOC2013B008.
References
[1] G. P. Agrawal, Nonlinear Fiber Optics, Academic Press, San Diego, 2012.10.1016/B978-0-12-397023-7.00011-5Search in Google Scholar
[2] A. I. Maimistov, Quantum Electron. 40, 756 (2010).10.1070/QE2010v040n09ABEH014396Search in Google Scholar
[3] Z. Y. Sun, Y. T. Gao, X. Yu, W. J. Liu, and Y. Liu, Phys. Rev. E 80, 066608 (2009).10.1103/PhysRevE.80.066608Search in Google Scholar
[4] M. M. Mani Rajan, A. Mahalingam, and A. Uthayakumar, Ann. Phys. 346, 1 (2014).10.1016/j.aop.2014.03.012Search in Google Scholar
[5] X. Y. Xie, B. Tian, W. R. Sun, Y. Sun, and D. Y. Liu, J. Mod. Opt. 62, 1374 (2015).10.1080/09500340.2015.1039944Search in Google Scholar
[6] G. I. Stegeman and M. Segev, Optical Spatial Solitons and Their Interactions: Universality and Diversity, Science 286, 1518 (1999).10.1126/science.286.5444.1518Search in Google Scholar
[7] S. Trillo and W. E. Torruellas, Spatial Solitons, Springer, Berlin, 2001.10.1007/978-3-540-44582-1Search in Google Scholar
[8] O. Cohen, R. Uzdin, T. Carmon, J. W. Fleischer, M. Segev, and S. Odoulov, Phys. Rev. Lett. 89, 133901 (2002).10.1103/PhysRevLett.89.133901Search in Google Scholar
[9] O. Lahav, H. Gurgov, P. Sidorenko, O. Peleg, L. Levi, et al. Opt. Lett. 37, 5196 (2012).10.1364/OL.37.005196Search in Google Scholar
[10] R. Wang, Z. Wu, Y. Zhang, Z. Zhang, C. Yuan, et al. Opt. Express 20, 14168 (2012).10.1364/OE.20.014168Search in Google Scholar
[11] Z. Li, L. Li, H. Tian, and G. Zhou, Phys. Rev. Lett. 84, 4096 (2000).10.1103/PhysRevLett.84.4096Search in Google Scholar
[12] K. Porsezian and K. Nakkeeran, Phys. Lett. A 206, 183 (1995).10.1016/0375-9601(95)00607-5Search in Google Scholar
[13] Y.S. Xue, B. Tian, W. B. Ai, M. Li, and P. Wang, Opt. Laser Technol. 48, 153 (2013).10.1016/j.optlastec.2012.09.026Search in Google Scholar
[14] D. W. Zuo, Y. T. Gao, L. Xue, Y. J. Feng, and Y. H. Sun, Appl. Math. Lett. 40, 78 (2015).10.1016/j.aml.2014.07.016Search in Google Scholar
[15] N. Aközbek, S. John, Phys. Rev. E 58, 3876 (1998).10.1103/PhysRevE.58.3876Search in Google Scholar
[16] S. L. McCall and E. L. Hahn, Phys. Rev. 183, 457 (1969).10.1103/PhysRev.183.457Search in Google Scholar
[17] M. Nakazawa, K. Suzuki, Y. Kimura, H. Kubota, Phys. Rev. A 45, R2682 (1992).10.1103/PhysRevA.45.R2682Search in Google Scholar
[18] G. Kurizki, D. Petrosyan, T. Opatrny, M. Blaauboer, and B. Malomed, J. Opt. Soc. Am. B 19, 2066 (2002).10.1364/JOSAB.19.002066Search in Google Scholar
[19] H. Y. Tseng and S. Chi, Phys. Rev. E 66, 056606 (2002).10.1103/PhysRevE.66.056606Search in Google Scholar PubMed
[20] S. John and V. I. Rupasov, Europhys. Lett. 46, 326 (1999).10.1209/epl/i1999-00264-2Search in Google Scholar
[21] B. Kalithasan, K. Porsezian, P. T. Dinda, and B.A. Malomed, J. Opt. A 11, 045205 (2009).10.1088/1464-4258/11/4/045205Search in Google Scholar
[22] A. I. Maimistov and A. M. Basharov, Nonlinear Optical Waves, Kluwer, London, 1999.10.1007/978-94-017-2448-7Search in Google Scholar
[23] A. I. Maimistov and E. A. Manykin, Sov. Phys. JETP 58, 685 (1983).Search in Google Scholar
[24] G. L. Lamb Jr, Elements of Soliton Theory, Wiley, New York, 1980.Search in Google Scholar
[25] K. Nakkeeran and K. Porsezian, J. Mod. Opt. 43, 693 (1996).10.1080/09500349608232776Search in Google Scholar
[26] K. Porsezian and K. Nakkeeran, Phys. Lett. A 206, 183 (1995).10.1016/0375-9601(95)00607-5Search in Google Scholar
[27] R. A. Vlasov and E.V. Doktorov, Inhomogeneous Optical Solitons in the Resonance Kerr-type Media. Dokl. Akad. Nauk BSSR 26, 322 (1982).Search in Google Scholar
[28] E. V. Doktorov and R. A. Vlasov, J. Mod. Opt. 30, 223 (1983).10.1080/713821151Search in Google Scholar
[29] A. I. Maimistov and E. A. Manykin, Sov. Phys. JETP 58, 685 (1983).10.1136/adc.58.9.685Search in Google Scholar
[30] S. Kakei and J. Satsuma, J. Phys. Soc. Jpn. 63, 885 (1994).10.1143/JPSJ.63.885Search in Google Scholar
[31] M. Nakazawa, E. Yamada, and H. Kubota, Phys. Rev. Lett. 66, 2625 (1991).10.1103/PhysRevLett.66.2625Search in Google Scholar
[32] K. Porsezian and K. Nakkeeran, J. Mod. Opt. 42, 1953 (1995).10.1080/09500349514551691Search in Google Scholar
[33] J. M. Dudley, C. Finot, D. J. Richardson, and G. Millot, Nat. Phys. 3, 597 (2007).10.1038/nphys705Search in Google Scholar
[34] A. C. Newell, J. Math. Phys. 19, 1126 (1978).10.1063/1.523759Search in Google Scholar
[35] V. N. Serkin and T. Y. L. Belyaeva, JETP Lett. 74, 573 (2001).10.1134/1.1455063Search in Google Scholar
[36] V. N. Serkin, V. M. Chapela, J. Percino, and T. L. Belyaeva, Opt. Commun. 192, 237 (2001).10.1016/S0030-4018(01)01216-0Search in Google Scholar
[37] M. M. Mani Rajan and A. Mahalingam, J. Math. Phys. 54, 043514 (2013).10.1063/1.4798477Search in Google Scholar
[38] M. M. Mani Rajan, A. Mahalingam, and A. Uthayakumar, J. Opt. 14, 105204 (2012).Search in Google Scholar
[39] S. Vijayalekshmi, M. M. Mani Rajan, A. Mahalingam, and A. Uthayakumar, J. Mod. Opt. 62, 278 (2015).10.1080/09500340.2014.975847Search in Google Scholar
[40] M. M. Mani Rajan, J. Hakkim, A. Mahalingam, and A. Uthayakumar, Eur. Phys. J. D 67, 150 (2013).10.1140/epjd/e2013-30748-7Search in Google Scholar
[41] M. M. Mani Rajan, A. Mahalingam, A. Uthayakumar, and K. Porsezian, Commun. Nonlinear Sci. Number. Simul. 18, 1410 (2013).10.1016/j.cnsns.2012.10.008Search in Google Scholar
[42] C. Q. Dai, Y. Y. Wang, Q. Tian, and J. F. Zhang, Ann. Phys. 327, 512 (2012).10.1016/j.aop.2011.11.016Search in Google Scholar
[43] H. P. Zhu, Nonlinear Dyn. 72, 873 (2013).10.1007/s11071-013-0759-1Search in Google Scholar
[44] S. Loomba, R, Gupta, K. K. De, C. N. Kumar, and T. S. Raju, Opt. Fiber Technol. 21, 20 (2015).10.1016/j.yofte.2014.07.006Search in Google Scholar
[45] N. Akhmediev, J. M. Soto-Crespo, J. M. and A. Ankiewicz, Phys. Lett. A 373, 2137 (2009).10.1016/j.physleta.2009.04.023Search in Google Scholar
[46] A. R. Osborne, Nonlinear Ocean Waves and the Inverse Scattering Transform. Academic Press, New York, 2010.10.1016/S0074-6142(10)97003-4Search in Google Scholar
[47] E. Pelinovsky and C. Kharif, Extreme ocean Waves, Springer, Berlin, 2008.10.1007/978-1-4020-8314-3Search in Google Scholar
[48] C. Garrett and J. Gemmrich, Phys. Today 62, 62 (2009).10.1063/1.3156339Search in Google Scholar
[49] W. R. Sun, B. Tian, H. L. Zhen, and Y. Sun, Nonlinear Dyn. 81, 725 (2015).10.1007/s11071-015-2022-4Search in Google Scholar
[50] D. R. Solli, C. Ropers, P. Koonath, and B. Jalali, Nature 450, 1054 (2007).10.1038/nature06402Search in Google Scholar PubMed
[51] W. R. Sun, B. Tian, Y. Jiang, and H. L. Zhen, Phys. Rev. E 91, 023205 (2015).10.1103/PhysRevE.91.023205Search in Google Scholar PubMed
[52] X. Y. Xie, B. Tian, W. R. Sun, and Y. Sun, Nonlinear Dyn. 81, 1349 (2015).10.1007/s11071-015-2073-6Search in Google Scholar
[53] M. M. Mani Rajan, A. Mahalingam, and A. Uthayakumar, J. Opt. 14, 105204 (2012).10.1088/2040-8978/14/10/105204Search in Google Scholar
[54] M. J. Ablowitz, D. J. Kaup, A. C. Newell, and H. Segur, Phys. Rev. Lett. 31, 125 (1973).10.1103/PhysRevLett.31.125Search in Google Scholar
[55] A. S. Fokas, Stud. Appl. Math 77, 253 (1987).10.1002/sapm1987773253Search in Google Scholar
[56] W. Hereman, Int. J. Quantum Chem. 106, 278 (2006).10.1002/qua.20727Search in Google Scholar
[57] M. Hisakado and M. Wadati, J. Phys. Soc. Jpn. 64, 408 (1995).10.1143/JPSJ.64.408Search in Google Scholar
[58] H. L. Zhen, B. Tian, Y. F. Wang, and D. Y. Liu, Phys. Plasmas 22, 032307 (2015).10.1063/1.4913668Search in Google Scholar
[59] H. L. Zhen, B. Tian, Y. Sun, and J. Chai, Phys. Plasmas 22, 102304 (2015).10.1063/1.4932076Search in Google Scholar
[60] B. Guo, L. Ling, and Q. P. Liu, Phys. Rev. E 85, 026607 (2012).10.1103/PhysRevE.85.026607Search in Google Scholar PubMed
[61] B. Yang, W. G. Zhang, H. Q. Zhang, and S. B. Pei, Phys. Scr. 88, 065004 (2013).10.1088/0031-8949/88/06/065004Search in Google Scholar
©2016 by De Gruyter
Articles in the same Issue
- Frontmatter
- Theoretical Investigations on the Elastic and Thermodynamic Properties of Rhenium Phosphide
- Lax Pair, Conservation Laws, Solitons, and Rogue Waves for a Generalised Nonlinear Schrödinger–Maxwell–Bloch System under the Nonlinear Tunneling Effect for an Inhomogeneous Erbium-Doped Silica Fibre
- Effect of Trace Fe3+ on Luminescent Properties of CaWO4: Pr3+ Phosphors
- Rogue-Wave Interaction of a Nonlinear Schrödinger Model for the Alpha Helical Protein
- Multi-Scale Long-Range Magnitude and Sign Correlations in Vertical Upward Oil–Gas–Water Three-Phase Flow
- Theoretical Study of Geometries, Stabilities, and Electronic Properties of Cationic (FeS)n+ (n = 1–5) Clusters
- Explanation of the Quantum-Mechanical Particle-Wave Duality through the Emission of Watt-Less Gravitational Waves by the Dirac Equation
- Closed Analytical Solutions of the D-Dimensional Schrödinger Equation with Deformed Woods–Saxon Potential Plus Double Ring-Shaped Potential
- Solitons, Bäcklund Transformation, Lax Pair, and Infinitely Many Conservation Law for a (2+1)-Dimensional Generalised Variable-Coefficient Shallow Water Wave Equation
- The Non-Alignment Stagnation-Point Flow Towards a Permeable Stretching/Shrinking Sheet in a Nanofluid Using Buongiorno’s Model: A Revised Model
- Rapid Communication
- Extrinsic and Intrinsic Contributions to Plasmon Peaks in Solids
Articles in the same Issue
- Frontmatter
- Theoretical Investigations on the Elastic and Thermodynamic Properties of Rhenium Phosphide
- Lax Pair, Conservation Laws, Solitons, and Rogue Waves for a Generalised Nonlinear Schrödinger–Maxwell–Bloch System under the Nonlinear Tunneling Effect for an Inhomogeneous Erbium-Doped Silica Fibre
- Effect of Trace Fe3+ on Luminescent Properties of CaWO4: Pr3+ Phosphors
- Rogue-Wave Interaction of a Nonlinear Schrödinger Model for the Alpha Helical Protein
- Multi-Scale Long-Range Magnitude and Sign Correlations in Vertical Upward Oil–Gas–Water Three-Phase Flow
- Theoretical Study of Geometries, Stabilities, and Electronic Properties of Cationic (FeS)n+ (n = 1–5) Clusters
- Explanation of the Quantum-Mechanical Particle-Wave Duality through the Emission of Watt-Less Gravitational Waves by the Dirac Equation
- Closed Analytical Solutions of the D-Dimensional Schrödinger Equation with Deformed Woods–Saxon Potential Plus Double Ring-Shaped Potential
- Solitons, Bäcklund Transformation, Lax Pair, and Infinitely Many Conservation Law for a (2+1)-Dimensional Generalised Variable-Coefficient Shallow Water Wave Equation
- The Non-Alignment Stagnation-Point Flow Towards a Permeable Stretching/Shrinking Sheet in a Nanofluid Using Buongiorno’s Model: A Revised Model
- Rapid Communication
- Extrinsic and Intrinsic Contributions to Plasmon Peaks in Solids
