Generalized Solution of the Third-Order Nonlinear Schrödinger Equation
https://doi.org/10.1134/S2304487X2101003X
Abstract
The generalized third order nonlinear Schrödinger equation is considered. This equation is a non-linear partial differential equation, which can be used to describe pulses in optical fibers. The Cauchy problem for it is not solved by the inverse scattering transform; for this reason, the solution of the equation is sought in the travelling wave variables. A system of differential equations for the imaginary and real parts has been obtained in these variables. Conditions for the existence of a solution of an overdetermined system of differential equations are determined. An analytical solution is found in terms of the Jacobi elliptic function. The solutions are presented in the form of periodic and solitary waves of the nonlinear Schrrödinger equation under various conditions on the coefficients. The found solutions are plotted in the graphical form.
About the Authors
E. V. Antonova
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. Kudryashov N. A. Metody` nelinejnoj matematicheskoj fiziki [Methods of nonlinear mathematical physics]. Dolgoprudny`j: Izd. Dom Intellekt, 2010. 364 p. (in Russian)
2. Uizem Dzh. Linejny`e i nelinejny`e volny` [Linear and nonlinear waves]. Moscow, Mir Publ., 1977. 622 p. (in Russian)
3. Zakharov V. E., Manakov S. V., Novikov S. P., Pitaevskij L. P. Teoriya solitonov: metod obratnoj zadachi [Theory of solitons: the method of the inverse problem]. 1980. 319 p. (in Russian)
4. Kudryashov N. A. Optical solitons of mathematical model with arbitrary refractive index // Optik. 2020. V. 224. P. 165391.
5. Kudryashov N. A. On traveling wave solutions of the Kundu–Eckhaus equation // Optik. 2020. V. 224. P. 165500.
6. Kudryashov N. A., Antonova E. V. Solitary waves of equation for propagation pulse with power nonlinearities // Optik. 2020. V. 217. P. 164881.
7. Zhou Q., Yao D., Dinga S., Zhang Y., Chen F., Chen F., Liu X., Spatial optical solitons in fifth order and seventh order weakly nonlocal nonlinear media // Optik. 2013. V. 124. P. 5683–5686.
8. Biswas A. 1-soliton solution of the generalized Radhakrishnan–Kundu–Laksmanan equation // Physics Letters A. 2009. V. 373. P. 2546–2548.
9. Biswas A. Optical soliton perturbation with Radhakrishnan-Kundu-Laksmanan equation by traveling wave hypothesis // Optik. 2018. V. 171. P. 217–220.
10. Triki H., Biswas A. Sub pico-second chirped envelope solitons and conservation laws in monomode optical fibers for a new derivative nonlinear Schrödinger’s model // Optics. 2018. V. 173. P. 235–241.
11. Zhou Q., Ekici M., Sonmezoglu A. Exact chirped singular soliton solutions of Triki – Biswas equation // Optik. 2019. V. 181. P. 338–342.
12. Nofal T. A., Zayed E. M. E., Alngar M. E. M., Shohib R. M. A., Ekici M. Highly dispersive optical solitons perturbation having Kudryashov’s arbitrary form with sextic-power law refractive index and generalized non-local laws // Optik. 2021. V. 228. P. 166120.
13. Yildirim Y., Biswas A., Ekici M., Triki H., Gonzalez-Gaxiola O., Alzahrani A. K., Belic M. R. Optical solitons in birefringent fibers for Radhakrishnan–Kundu–Lakshmanan equation with five prolific integration norms // Optik. 2020. V. 208. P. 164550.
14. Kohl R. W., Biswas A., Zhou Q., Ekici M., Alzahrani A. K., Belic M. R. Optical soliton perturbation with polynomial and triple-power laws of refractive index by semi-inverse variational principle // Optik. 2020. V. 135. P. 109765.
15. Ekici M., Sonmezoglu A. Optical solitons with Biswas–Arshed equation by extended trial function method // Optik. 2019. V. 177. P. 13–20.
16. Zayed E. M. E., Alngar M. E. M. Optical soliton solutions for the generalized Kudryashov equation of propagation pulse in optical fiber with power nonlinearities by three integration algorithms // Math. Methods Appl. Sci. 2021. V. 44 (1). P. 315–324.
17. Kudryashov N. A. First integral and general solution of traveling wave reduction for the Triki–Biswas equation // Optik. 2019. V. 185. P. 275–281.
18. Kudryashov N. A. First integrals and general solution of the traveling wave reduction for Schrëdinger equation with anti-cubic nonlinearity // Optik. 2019. V. 185. P. 665–671.
19. Kudryashov N. A. Exact solutions of the equation for surface waves in a convecting fluid // Appl. Math. Comput. 2019. V. 344. P. 97–106.
20. Kudryashov N. A. Traveling wave solutions of the generalized nonlinear Schrëdinger equation with cubic-quintic nonlinearity // Optik. 2019. V. 188. P. 27–35.
21. Kudryashov N. A. A generalized model for description pulses in optical fiber // Optik. 2019. V. 189. P. 42–52.
22. Kudryashov N. A. Method for finding highly dispersive optical solitons of nonlinear differential equations // Optik. 2020. V. 212. P. 163550.
23. Kudryashov N. A. Solitary wave solutions of hierarchy with non-local nonlinearity // Appl. Math. Lett. 2020. V. 103. P. 106155.
24. Kudryashov N. A. Highly dispersive solitary wave solutions of perturbed nonlinear Schrëdinger equations // Appl. Math. Comput. 2020. V. 371. P. 124972.
25. Kudryashov N. A. Mathematical model of propagation pulse in optical fiber with power nonlinearities // Optik. 2020. V. 212. P. 164750.
26. Dan J., Sain S., Ghose-Choudhary A., Garai S. Application of the Kudryashov function for finding solitary wave solutions of NLS type differential equations // Optik. 2020. V. 224. P. 165519.
27. Kudryashov N. A. Construction of nonlinear differential equations for description of propagation pulses in optical fiber // Optik. 2019. V. 192. P. 162964.
Review
For citations:
Antonova E.V., Kudryashov N.A. Generalized Solution of the Third-Order Nonlinear Schrödinger Equation. Vestnik natsional'nogo issledovatel'skogo yadernogo universiteta "MIFI". 2021;10(1):19-26. (In Russ.) https://doi.org/10.1134/S2304487X2101003X
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