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Zh. Vychisl. Mat. Mat. Fiz., 2009, Volume 49, Number 11, Pages 1988–2000 (Mi zvmmf4784)  

This article is cited in 6 scientific papers (total in 6 papers)

Numerical method for finding 3D solitons of the nonlinear Schrödinger equation in the axially symmetric case

O. V. Matusevich, V. A. Trofimov

Faculty of Computational Mathematics and Cybernetics, Moscow State University, Moscow, 119992, Russia

Abstract: A system of two nonlinear Schrödinger equations is considered that governs the frequency doubling of femtosecond pulses propagating in an axially symmetric medium with quadratic and cubic nonlinearity. A numerical method is proposed to find soliton solutions of the problem, which is previously reformulated as an eigenvalue problem. The practically important special case of a single Schrödinger equation is discussed. Since three-dimensional solitons in the case of cubic nonlinearity are unstable with respect to small perturbations in their shape, a stabilization method is proposed based on weak modulations of the cubic nonlinearity coefficient and variations in the length of the focalizing layers. It should be emphasized that, according to the literature, stabilization was previously achieved by alternating layers with oppositely signed nonlinearities or by using nonlinear layers with strongly varying nonlinearities (of the same sign). In the case under study, it is shown that weak modulation leads to an increase in the length of the medium by more than 4 times without light wave collapse. To find the eigenfunctions and eigenvalues of the nonlinear problem, an efficient iterative process is constructed that produces three-dimensional solitons on large grids.

Key words: nonlinear Schrödinger equations, three-dimensional solitons, numerical method for computing eigenvalues and eigenfunctions, iterative process.

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English version:
Computational Mathematics and Mathematical Physics, 2009, 49:11, 1902–1912

Bibliographic databases:

UDC: 519.634
Received: 06.03.2009

Citation: O. V. Matusevich, V. A. Trofimov, “Numerical method for finding 3D solitons of the nonlinear Schrödinger equation in the axially symmetric case”, Zh. Vychisl. Mat. Mat. Fiz., 49:11 (2009), 1988–2000; Comput. Math. Math. Phys., 49:11 (2009), 1902–1912

Citation in format AMSBIB
\by O.~V.~Matusevich, V.~A.~Trofimov
\paper Numerical method for finding 3D solitons of the nonlinear Schr\"odinger equation in the axially symmetric case
\jour Zh. Vychisl. Mat. Mat. Fiz.
\yr 2009
\vol 49
\issue 11
\pages 1988--2000
\jour Comput. Math. Math. Phys.
\yr 2009
\vol 49
\issue 11
\pages 1902--1912

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    This publication is cited in the following articles:
    1. Trofimov V.A., Matusevich O.V., Smotrov D.A., “Mode of propagation of optical radiation with self-similar pulse shape in layered medium with nonlinear absorption”, Active Photonic Materials IV, Proceedings of SPIE, 8095, 2011  isi
    2. Trofimov V.A., Zakharova I.G., Smotrov D.A., Lan Sh., “Self-Similar Pulse Shape Mode for Femtosecond Pulse Propagation in Optical Fiber with Multi-Photon Absorption and Nonlinear Refraction”, Micro-Structured and Specialty Optical Fibres II, Proceedings of SPIE, 8775, eds. Kalli K., Kanka J., Mendez A., SPIE-Int Soc Optical Engineering, 2013  crossref  isi  scopus
    3. Savenkova N.P., Laponin V.S., “Chislennoe issledovanie metodov poiska mnogomernykh solitonov”, Vestnik Sibirskogo gosudarstvennogo aerokosmicheskogo universiteta im. akademika M.F. Reshetneva, 2013, no. 2(48), 81–85  mathscinet  elib
    4. Trofimov V.A. Zakharova I.G. Konar S., “Self-Similar Pulse-Shape Mode For Femtosecond Pulse Propagation in Medium With Resonant Nonlinearity”, Nonlinear Optics and Its Applications VIII; and Quantum Optics III, Proceedings of Spie, 9136, ed. Eggleton B. Gaeta A. Broderick N. Sergienko A. Rauschenbeutel A. Durt T., Spie-Int Soc Optical Engineering, 2014, 91360Z  crossref  isi  scopus
    5. Trofimov V.A. Zakharova I.G. Fedotov M.V., “Self-Similar Shape Mode of Optical Pulse Propagation in de-Focusing Medium With Two-Photon Absorption”, 22nd International Laser Physics Workshop, Journal of Physics Conference Series, 497, IOP Publishing Ltd, 2014, 012023  crossref  isi  scopus
    6. Trofimov V.A. Zakharova I.G., “Propagation of Femtosecond Pulse With Self-Similar Shape in Medium With Non-Linear Absorption”, Nonlinear Optics and Applications Ix, Proceedings of Spie, 9503, ed. Bertolotti M. Haus J. Zheltikov A., Spie-Int Soc Optical Engineering, 2015, 95030R  crossref  isi  scopus
  • Журнал вычислительной математики и математической физики Computational Mathematics and Mathematical Physics
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