Stability of Optical Pulses Described by the Perturbed Ginzburg–Landau Equation
https://doi.org/10.56304/S2304487X22030075
Abstract
A perturbed second-order ordinary differential equation obtained by passing to traveling wave variables in the generalized Ginzburg–Landau equation is considered. The stability of the stationary points of the equation is analyzed and intervals of parameters are found for which the system has separatrices of saddle points. Explicit expressions for the homoclinic and heteroclinic orbits of the system are found for two special cases of parameters. The stability of these orbits is analyzed t by constructing the Melnikov function along them. The analysis of the zeros of the Melnikov function allows us to determine the ranges of the control parameters of the system where the necessary condition for the occurrence of Melnikov chaos is satisfied.
About the Authors
S. F. Lavrova
National Research Nuclear University MEPhI (Moscow Engineering Physics Institute)
Russian Federation
Russian Federation
Moscow
115409
N. A. Kudryashov
National Research Nuclear University MEPhI (Moscow Engineering Physics Institute)
Russian Federation
Russian Federation
Moscow
115409
References
1. Agarwal P., Deniz S., Jain S., Alderremy A.A., Aly S. A new analysis of a partial differential equation arising in biology and population genetics via semi analytical techniques. Physica A: Statistical Mechanics and its Applications, 2020, vol. 542, Article ID 122769.
2. FitzHugh R. Mathematical models of threshold phenomena in the nerve membrane. The bulletin of mathematical biophysics, 1955, vol. 17, no. 4, pp. 257–278.
3. Grinfeld M.A. The stress driven instability in elastic crystals: Mathematical models and physical manifestations. Journal of Nonlinear Science, 1993, vol. 3, no. 1, pp. 35–83.
4. Lumley J.L. Computational modeling of turbulent flows. Advances in applied mechanics, 1979, vol. 18, pp. 123–176.
5. Bec J., Khanin K. Burgers turbulence. Physics reports, 2007, vol. 447, no. 1–2, pp. 1–66.
6. Holmes P.J. Bifurcations to divergence and flutter in flow-induced oscillations: a finite dimensional analysis. Journal of Sound and Vibration, 1977, vol. 53, no. 4, pp. 471–503.
7. Kudryashov N.A. First integrals and general solution of the complex Ginzburg-Landau equation. Applied Mathematics and Computation, 2020, vol. 386, Article ID 125407.
8. Biswas A. Temporal 1-soliton solution of the complex Ginzburg-Landau equation with power law nonlinearity. Progress in Electromagnetics Research, 2009, vol. 96, pp. 1–7.
9. Shwetanshumala S. Temporal solitons of modified complex Ginzberg Landau equation. Progress In Electromagnetics Research Letters, 2008, vol. 3, pp. 17–24.
10. Zayed E.M., Nofal T.A., Alngar M.E., El-Horbaty M.M. Cubic-quartic optical soliton perturbation in polarization- preserving fibers with complex Ginzburg- Landau equation having five nonlinear refractive index structures. Optik, 2021, vol. 231, Article ID 166381.
11. Gepreel K.A., Zayed E.M.E., Alngar M.E.M. New optical solitons perturbation in the birefringent fibers for the CGL equation with Kerr law nonlinearity using two integral schemes methods. Optik, 2021, vol. 227, pp. 166099.
12. Melnikov V.K. Ob ustojchivosti centra pri periodicheskih po vremeni vozmushcheniyah. Trudy Moskovskogo matematiches kogo obshchestva, 1963, vol. 12, pp. 3–52. (in Russian)
13. Holmes C., Holmes P. Second order averaging and bifurcations to subharmonics in Duffing’s equation. Journal of Sound and Vibration, 1981, vol. 78, no. 2, pp. 161–174.
14. Guckenheimer J., Holmes P. Nonlinear oscillations, dynamical systems, and bifurcations of vector fields. Springer Science & Business Media, 2013.
15. Kudryashov N.A., Lavrova S.F. Complex dynamics of perturbed solitary waves in a nonlinear saturable medium: A Melnikov approach. Optik, 2022, vol. 265, Article ID 169454.
16. Kudryashov N.A., Lavrova S.F. 2021. Dynamical properties of the generalized model for description of propagation pulses in optical fiber with arbitrary refractive index. Optik, 2022, vol. 245, Article ID 167679.
17. Lavrova S.F., Kudryashov N.A. Metod Mel’nikova dlya obobshchennogo uravneniya Duffinga. [Melnikov’s method for the generalized Duffing equation]. Vestnik NIYaU MIFI, 2021, vol. 10, no. 2. pp. 135–142. (in Russian)
18. Lavrova S.F., Kudryashov N.A. Primenenie metoda Mel’nikova k uravneniyu Triki–Bisvasa. [Application of the Melnikov Method to the Tricky–Biswas Equation]. Vestnik NIYaU MIFI, 2021, vol. 10, no. 4, pp. 308–317. (in Russian)
19. Davis H.T. Introduction to nonlinear differential and integral equations. US Atomic Energy Commission, 1960.
20. Kudryashov N.A. Solitary and periodic waves of the hierarchy for propagation pulse in optical fiber. Optik, 2019, vol. 194, Article ID 163060.
21. Zaitsev V.F., Polyanin A.D. Handbook of exact solutions for ordinary differential equations. CRC press, 2002.
Review
For citations:
Lavrova S.F., Kudryashov N.A. Stability of Optical Pulses Described by the Perturbed Ginzburg–Landau Equation. Vestnik natsional'nogo issledovatel'skogo yadernogo universiteta "MIFI". 2022;11(3):235–247. (In Russ.) https://doi.org/10.56304/S2304487X22030075
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