Nonlinear Dynamic Processes Described by the Radhakrishnan–Kundu–Lakshmanan Equations
https://doi.org/10.1134/S2304487X20010058
Abstract
The propagation of solitary waves through optical media is described by a variety of nonlinear partial differential equations. In this work, the nonlinear dynamics of dispersive nonlinear waves in polarization–preserving fibers having Kerr nonlinearity has been described by the Radhakrishnan–Kundu–Lakshmanan equation. Dimensionless variables are used in the initial equation to study of dynamic processes. The introduction of the traveling wave variables reduces the equation to the system of two third order ordinary differential equations. The divergence is calculated for the normal form vector field of this system. It has been found that this system of equations is not dissipative. The largest Lyapunov exponents are computed by the Bennetin algorithm for the different values of one of the model parameters. Despite the presence of a positive largest Lyapunov exponent for some parameter values, there are no attractors or chaotic dynamical regimes in this system, since the solution unboundedly decreases in one variable and the initial point in the algorithm has to be chosen on the attractor.
Supporting agencies: The study was supported by a grant from the Russian Science Foundation (project 18-11- 00209)
About the Authors
S. F. Lavrova
National Research Nuclear University MEPhI (Moscow Engineering Physics Institute)
Russian Federation
Russian Federation
115409
Moscow
N. A. Kudryashov
National Research Nuclear University MEPhI (Moscow Engineering Physics Institute)
Russian Federation
Russian Federation
115409
Moscow
References
1. Biswas Anjan. “Optical solitons: Quasi-stationarity versus Lie transform.” Optical and Quantum Electronics. 2003. 35.10. P. 979–998.
2. Khalique C. M., Biswas A. “Optical solitons with parabolic and dual-power law nonlinearity via Lie symmetry analysis.” Journal of Electromagnetic Waves and Applications. 2009. 23.7. P. 963–973.
3. Kohl Russell et al. “Optical soliton perturbation in a non-Kerr law media.” Optics & Laser Technology. 2008. 40.4. P. 647–662.
4. Topkara Engin et al. “Optical solitons with non-Kerr law nonlinearity and inter-modal dispersion with time-dependent coefficients.” Communications in Nonlinear Science and Numerical Simulation. 2010. 15. 9. P. 2320–2330.
5. Konar S., Mishra M., Soumendu Jana. “Nonlinear evolution of cosh-Gaussian laser beams and generation of flat top spatial solitons in cubic quintic nonlinear media.” Physics Letters A. 2007. 362.5-6. P. 505–510.
6. Yan Zhenya. “Generalized method and its application in the higher-order nonlinear Schrodinger equation in nonlinear optical fibres.” Chaos, Solitons & Fractals. 2003. 16. 5. P. 759–766.
7. Triki Houria, and Abdul-Majid Wazwaz. “Combined optical solitary waves of the Fokas—Lenells equation.” Waves in Random and Complex Media. 2017. 27. 4. P. 587–593.
8. Biswas A. “1-soliton solution of the generalized Radhakrishnan, Kundu, Lakshmanan equation.” Physics Letters A. 2009. 373.30. P. 2546–2548.
9. Zhang Jianming, Shuming Li, and Hongpeng Geng. “Bifurcations of exact travelling wave solutions for the generalized RKL equation.” J. Appl. Anal. Comput. 2016. 6.4. P. 1205–1210.
10. Biswas A. Optical soliton perturbation with Radhakrishnan–Kundu–Laksmanan equation by traveling wave hypothesis, Optik. 2018. V. 171. P. 217–220.
11. Biswas A., Ekici M., Sonmezoglu A., Alshomrani A. S., Optical soliton with Radhakrishnan–Kundu–Laksmanan by extended trial function scheme, Optik. 2018. V. 160. P. 415–427.
12. Gonzalez-Gaxiola O., Anjan Biswas, Optical solitons with Radhakrishnan–Kundu–Laksmanan equation by Laplace–Adomian decomposition method, Optik. 2019. V. 179. 434–442.
13. Kydryashov N. A., Safonova D. V., A. Biswas, Painleve analysis and solution to travelling wave reduction of Radhakrishnan-Kundu-Lakshmanan equation, Regular and Chaotic Dynamics. 2019. 24.6. P. 607–614.
14. Lu Dianchen, Aly R. Seadawy, and Mostafa M. A. Khater. “Dispersive optical soliton solutions of the generalized Radhakrishnan–Kundu–Lakshmanan dynamical equation with power law nonlinearity and its applications.” Optik. 2018. 164. P. 54–64.
15. Benettin Giancarlo et al. “Lyapunov characteristic exponents for smooth dynamical systems and for Hamiltonian systems; a method for computing all of them. Part 1: Theory.” Meccanica. 1980. 15. 1. P. 9–20.
Review
For citations:
Lavrova S.F., Kudryashov N.A. Nonlinear Dynamic Processes Described by the Radhakrishnan–Kundu–Lakshmanan Equations. Vestnik natsional'nogo issledovatel'skogo yadernogo universiteta "MIFI". 2020;9(1):45-49. (In Russ.) https://doi.org/10.1134/S2304487X20010058
Views:
102
Email this article 