Bifurcation Analysis of Cubic Type Nonlinear Schrödinger Equation with Dispersion and Attenuation in Optical Fiber
DOI:
https://doi.org/10.30829/zero.v9i2.25440Keywords:
Attenuation, Bifurcation, Dispersion, Nonlinear Schrödinger equation, Nonlinearity, Optical fibers.Abstract
This study investigates the bifurcation of the cubic Nonlinear Schrödinger Equation (NLSE) in optical fiber media by considering dispersion and attenuation effects. The NLSE models light pulse propagation, with γ representing the strength of nonlinearity. Analytical derivations yield stationary solutions, while numerical simulations using the Newton-Raphson method and eigenvalue analysis verify stability. Results show a critical bifurcation at γ=0: for γ ≤ 0, the system exhibits one unstable fixed point, whereas for γ > 0, three fixed points appear, with simulations confirming that only the two nontrivial branches are stable. This corresponds to a pitchfork bifurcation and a stability transition governed primarily by nonlinearity. Although attenuation is included in the model, its contribution is negligible, indicating that bifurcation behavior is dominated by γ. Compared with previous studies focusing on dispersion-only NLSE or fractional/extended models, this work highlights the decisive role of nonlinearity in determining fixed points and stability in optical fibers.References
M. Ramli, M. Ikhwan, N. Nazaruddin, H. A. Mardi, T. Usman, and E. Safitri, "Multi-peak soliton dynamics and decoherence via the attenuation effects and trapping potential based on a fractional nonlinear Schrödinger cubic quintic equation in an optical fiber," Alexandria Eng. J., vol. 107, no. November 2023, pp. 507-520, 2024, doi: 10.1016/j.aej.2024.07.037.
H. Abdillah Mardi, N. Nasaruddin, M. Ikhwan, N. Nurmaulidar, and M. Ramli, "Soliton dynamics in optical fiber based on nonlinear Schrödinger equation," Heliyon, vol. 9, no. 3, 2023, doi: 10.1016/j.heliyon.2023.e14235.
M. F. Alotaibi, N. Raza, M. H. Rafiq, and A. Soltani, "New solitary waves, bifurcation and chaotic patterns of Fokas system arising in monomode fiber communication system," Alexandria Eng. J., vol. 67, pp. 583-595, 2023, doi: 10.1016/j.aej.2022.12.069.
J. S. Russel, "The report of the meeting of the British Association for the advancement of science [5]," 1870. doi: 10.1038/002124a0.
D. Fadhiliani et al., "The dynamics of surface wave propagation based on the Benjamin Bona Mahony equation," Heliyon, vol. 6, no. 5, p. e04004, 2020, doi: 10.1016/j.heliyon.2020.e04004.
D. J. Zhang, S. L. Zhao, Y. Y. Sun, and J. Zhou, "Solutions to the modified Korteweg-de Vries equation," Rev. Math. Phys., vol. 26, no. 7, pp. 1-42, 2014, doi: 10.1142/S0129055X14300064.
G. P. Agrawal, "Chapter 5 - Optical Solitons," in Nonlinear Fiber Optics (Fifth Edition), Fifth Edition., G. Agrawal, Ed., in Optics and Photonics. , Boston: Academic Press, 2013, pp. 129-191. doi: https://doi.org/10.1016/B978-0-12-397023-7.00005-X.
R. Boyd and B. Masters, "Nonlinear Optics, Third Edition," J. Biomed. Opt., vol. 14, p. 29902, 2009, doi: 10.1117/1.3115345.
L. Hakim and A. Kusumastuti, "Generalisasi Fungsi Airy sebagai Solusi Analitik Persamaan Schrodinger Nonlinier," CAUCHY J. Mat. Murni dan Apl., vol. 2, no. 2, pp. 86-95, 2012, doi: 10.18860/ca.v2i2.2223.
G. Agrawal, "Nonlinear Fiber Optics," Nonlinear Fiber Opt., 2006, doi: 10.1016/B978-0-12-369516-1.X5000-6.
A. Choudhuri and K. Porsezian, "Impact of dispersion and non-Kerr nonlinearity on the modulational instability of the higher-order nonlinear Schrödinger equation," Phys. Rev. A - At. Mol. Opt. Phys., vol. 85, no. 3, pp. 1-6, 2012, doi: 10.1103/PhysRevA.85.033820.
R. Fukuizumi and A. Sacchetti, "Bifurcation and Stability for Nonlinear Schrödinger Equations with Double Well Potential in the Semiclassical Limit," J. Stat. Phys., vol. 145, no. 6, pp. 1546-1594, 2011, doi: 10.1007/s10955-011-0356-y.
H. Qausar, M. Ramli, S. Munzir, M. Syafwan, H. Susanto, and V. Halfiani, "Nontrivial on-site soliton solutions for stationary cubic-quintic discrete nonlinear schrodinger equation," IAENG Int. J. Appl. Math., vol. 50, no. 2, pp. 1-5, 2020.
S. J. Chapman, M. Kavousanakis, E. G. Charalampidis, I. G. Kevrekidis, and P. G. Kevrekidis, "A spectral analysis of the nonlinear Schrödinger equation in the co-exploding frame," Phys. D Nonlinear Phenom., vol. 439, 2022, doi: 10.1016/j.physd.2022.133396.
M. Zhen and B. Zhang, "Normalized ground states for the critical fractional NLS equation with a perturbation," Rev. Mat. Complut., vol. 35, no. 1, pp. 89-132, 2022, doi: 10.1007/s13163-021-00388-w.
Y. Y. Yan and W. J. Liu, "Soliton Rectangular Pulses and Bound States in a Dissipative System Modeled by the Variable-Coefficients Complex Cubic-Quintic Ginzburg-Landau Equation," Chinese Phys. Lett., vol. 38, no. 9, pp. 1-4, 2021, doi: 10.1088/0256-307X/38/9/094201.
B. S. T. Alkahtani, "Propagation of wave insights to the Chiral Schrödinger equation along with bifurcation analysis and diverse optical soliton solutions," Alexandria Eng. J., vol. 108, pp. 800-810, 2024, doi: 10.1016/j.aej.2024.08.116.
D. Damanik, M. Ruzhansky, V. Vougalter, M. W. Wong, R. Adami, and D. Noja, "Exactly solvable models and bifurcations: The case of the cubic NLS with a or a interaction in dimension one," Math. Model. Nat. Phenom., vol. 9, no. 5, pp. 1-16, 2014, doi: 10.1051/mmnp/20149501.
J. Song, B. A. Malomed, and Z. Yan, "Symmetry-breaking bifurcations and excitations of solitons in linearly coupled NLS equations with P T -symmetric potentials," Proc. R. Soc. A Math. Phys. Eng. Sci., vol. 479, no. 2278, pp. 1-20, 2023, doi: 10.1098/rspa.2023.0457.
A. M. Djaouti, M. Roshid, and H. Roshid, "Optimizing Optical Fiber Communications : Bifurcation Analysis and Soliton Dynamics in the Quintic Kundu - Eckhaus Model," pp. 1-24, 2025.
Alligood, Kathleen T, T. D. Sauer, and J. A. Yorke, Introduction to Dynamical Systems. United State of America: Springer-Verlag New York, Inc, 2000. doi: 10.1201/b16036-24.
C. J. Pethick and H. Smith, Bose-Einstein Condensation in Dilute Gases, 2nd ed. Cambridge University Press, 2008.
A. Walia and B. Singh, "Solution of Non-linear Equations Using Newton Raphson and Quasi-Newton Method and Application in Engineering Field. AIJR Abstracts," vol. 22, no. 4, pp. 2164-2181, 2022.
N. İ. Karabaş, S. Ö. Korkut, G. Tanoğlu, I. Aziz, and Siraj-ul-Islam, "An efficient approach for solving nonlinear multidimensional Schrödinger equations," Eng. Anal. Bound. Elem., vol. 132, no. August, pp. 263-270, 2021, doi: 10.1016/j.enganabound.2021.07.009.
S. Strogatz author, Nonlinear dynamics and chaos : with applications to physics, biology, chemistry, and engineering. Second edition. Boulder, CO : Westview Press, a member of the Perseus Books Group, [2015]. [Online]. Available: https://search.library.wisc.edu/catalog/9910223127702121
S. Wiggins, Introduction to applied nonlinear dynamical systems and chaos, vol. 75, no. 472. springer, 2003. doi: 10.2307/3620310.
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