Exact Solutions of One of the Nonlinear Differential Equations in Nonlinear Optics
https://doi.org/10.1134/S2304487X21050047
Abstract
One of the nonlinear partial differential equations used to describe the propagation of pulses in nonlinear optics is considered. The Cauchy problem for this equation cannot be solved by the method of the inverse scattering problem; for this reason, traveling wave variables are used to find the exact solution of the nonlinear second-order partial differential equation. It is proved that the considered equation has exact solutions in the form of periodic and solitary waves, which are determined using elliptic functions. The values of the parameters for the existence of exact solutions of this equation are given. Exact solutions of differential equations are expressed in terms of the Weierstrass elliptic function. A formula for describing solitary waves is also given. Exact solutions in the form of periodic and solitary waves are illustrated. It is shown that the nonlinearity index has a significant effect on the shape of a pulse propagating in an optical fiber.
Keywords
nonlinear differential equation,
traveling wave,
Weierstrass elliptic function,
optical fiber,
pulse propagation
Supporting agencies: The work was supported by the Ministry of Science and Higher Education of the Russian Federation (project state order No. 0723-2020-0036)
About the Authors
N. V. Ermolaeva
National Research Nuclear University MEPhI (Moscow Engineering Physics Institute); Volgodonsk Engineering Technical Institute, National Research Nuclear University MEPhI (Moscow Engineering Physics Institute)
Russian Federation
Russian Federation
115409
347360
Moscow
Rostovskaya oblast
Volgodonsk
N. A. Kudryashov
National Research Nuclear University MEPhI (Moscow Engineering Physics Institute)
Russian Federation
Russian Federation
115409
Moscow
References
1. Kudryashov N. A. Solitary and periodic waves of the hierarchy for propagation pulse in optical fiber // Optik, 2019. V. 194. 163060.
2. Biswas A., Vega-Guzman J., Mahmood M. F., Khan S., Ekii M., Zhou Q., Moshokoa S. P., Belik M. R. Solitons in optical fiber Bragg gratings with dispersive reflectivity // Optik. 2019. V. 182. P. 119–123.
3. Biswas A., Ekici M., Sonmezoglu A., Belik M. R. Solitons in optical fiber Bragg gratings with dispersive reflectivity by extended trial function method // Optik, 2019. V. 182. P. 88–94.
4. Polyanin A. D., Zaitsev V. F. Handbook of Exact Solutions for Ordinary Differential Equations. Boca Ration-New York, Chapman and Hall / CRC Press, 2003. 814 p.
5. Kudryashov N. A. Traveling wave reduction for the modified KdV hierarchy: Lax pair and first integrals // Communications in Nonlinear Science and Numerical Simulat, 2019. V. 73. P. 472–480.
6. Kudryashov N. A. General solution of the traveling wave reduction for the Kundu-Mukherjee-Naskar model // Optik, 2019. V. 186. P. 22–27.
7. Kudryashov N. A. General solution of the traveling wave reduction for the Chen-Lee-Liu equation // Optik, 2019. V. 186. P. 339–349.
Review
For citations:
Ermolaeva N.V., Kudryashov N.A. Exact Solutions of One of the Nonlinear Differential Equations in Nonlinear Optics. Vestnik natsional'nogo issledovatel'skogo yadernogo universiteta "MIFI". 2021;10(5):407-411. (In Russ.) https://doi.org/10.1134/S2304487X21050047
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