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Schrödinger Equation with a Cubic Nonlinearity Sech-Shaped Soliton Solutions

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DOI: 10.4236/opj.2012.23026    5,861 Downloads   9,014 Views   Citations
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ABSTRACT

We first analyze the sech-shaped soliton solutions, either spatial or temporal of the 1D-Schr?dinger equation with a cubic nonlinearity. Afterwards, these solutions are generalized to the 2D-Schr?dinger equation in the same configuration and new soliton solutions are obtained. It is shown that working with dimensionless equations makes easy this generalization. The impact of solitons on modern technology is then stressed.

Conflicts of Interest

The authors declare no conflicts of interest.

Cite this paper

P. Hillion, "Schrödinger Equation with a Cubic Nonlinearity Sech-Shaped Soliton Solutions," Optics and Photonics Journal, Vol. 2 No. 3, 2012, pp. 173-177. doi: 10.4236/opj.2012.23026.

References

[1] T. Oh and C. Sulem, “On One Dimensional Cubic Nonlinear Schr?dinger Equation,” arXiv : 10072018, 2010.
[2] N. N. Akhmediev, “Spatial Solitons in Kerr and KerrLike Media,” Optical and Quantum Electronics, Vol. 30, No. 7-10, 1998, pp. 535-569. doi:10.1023/A:1006902715737
[3] J. G. New, “Introduction to Nonlinear Optics,” Cambridge University Press, Cambridge, 2011. doi:10.1017/CBO9780511975851
[4] G. Gryndberg, A. Aspect and Cl. Fabre, “Introduction to Quantum Optics,” Cambridge University Press, Cambridge, 2010.
[5] C. Q. Dai, Z. Q. Qin and C. L. Zheng, “Multisoliton Solutions of the Modified Nonlinear Schr?-Dinger Equation,” Physica Scripta, Vol. 85, No. 4, 2012, p. 045007. doi:10.1088/0031-8949/85/04/045007
[6] C. J. Brenton, “Solitons and Nonlinear Optics in SiliconOn Insulator Photonics Wires,” Thesis, 2009.
[7] G. B. Whitham, “Linear and Nonlinear Wave Equations,” Wiley, New York, 1974.
[8] Eq. World (Google), “Shr?dinger Equation with a Cubic Nonlinearity,”.
[9] J. Bourgain, “Nonlinear Schr?dinger Equation,” Parc City Lectures, 1995.
[10] R. Killip, T. Tao and M. Visam, “The Cubic Nonlinear Schr?dinger Equation in Two Dimen-Sions with Radial Data,” Journal of the European Mathematical Society, Vol. 11, No. 6, 2009, pp. 1203-1258. doi:10.4171/JEMS/180
[11] A. R. Seadway, “Exact Solutions of a Two Dimensional Nonlinear Schr?dinger Equation,” Applied Mathematics Letters, Vol. 25, 2012, pp. 687-691.
[12] J. C. Bronski, M. Segev and M. Weinstein, “Mathematical Frontiers in Optical Solitons,” Proceedings of the National Academy of Sciences, Vol. 98, No. 23, 2001, pp. 12872-12873.
[13] Z. H. Muslinami, K. G. Makris, R. EL-Ganaini and D. N. Christodoulides, “Optical Solitons in PT Periodic Potentials,” Physical Review Letters, Vol. 100, 2006, p. 030402.
[14] Y. V. Kartshov, “Optical Lattice Solitons: Guiding and Routing Light at Will,” OSA Optics and Photonics Focus, 2009.
[15] F. I. Khatri, “Optical Soliton Propagation and Control,” MIT Thesis, Massachusetts Institute of Technology, Cambridge, 1996.
[16] Y. V. Kartshov, J. A. Malomed, V. A. Vysloukh and L. Torner, “Two Dimensional Solitons in Nonlinear Lattices,” Optics Letters, Vol. 34, No. 6, 2009, pp. 770-777. doi:10.1364/OL.34.000770
[17] M. J. Ablowitz, B. Ilan, E. Schonbrun and R. Piestun, “Two Dimensional Solitons in Irregular Lattices” Theoretical and Mathematical Physics, Vol. 151, No. 3, 2007, pp. 723-734. doi:10.1007/s11232-007-0058-4
[18] G. Maugin, “Nonlinear Surface Wave and Solitons,” The European Physical Journal, Vol. 147, No. 1, 2007, pp. 209-230.
[19] X. Wang, A. Bezryadina, Z. ChenK, G. Magris, D. N. Christodoulides and G. I. Stegman, “Observation of a Two Dimensional Surface Soliton,” Physical Review, Vol. 98, 2007, p. 123903.
[20] Y. Kominis, A. Papadopoulos and K. Hizanidis, “Surface Solitons in Wave Guide Arrays; Analytical Solutions,” Optics Express, Vol. 15, No. 16, 2007, pp. 10041-10051. doi:10.1364/OE.15.010041
[21] Y. V. Kartshov, V. A. Vysloukh and L. Torner, “Generation of Surface Soliton Arrays,” Optics Letters, Vol. 31, No. 15, 2006, pp. 2329-2331. doi:10.1364/OL.31.002329
[22] N. K. Efremides, J. Hudlock, D. N. Christodoulides, J. W. Fleischer, D. Cohen and M. Segev, “Two Dimensional Optical Lattice Solitons,” Physical Review Letters, Vol. 91, No. 21, 2003, pp. 1-4.
[23] B. A. Malomed, D. Mialache, F. Wise and L. Torner, “Spatial Temporel Solitons,” Journal of Optics B: Quantum and Semiclassical Optics, Vol. 7, No. 5, 2005, pp. R53-R72. doi:10.1088/1464-4266/7/5/R02

  
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