Chebyshev spectral method for numerical simulations of counter-propagating optical waves interaction in nonlinear media

Authors

  • Yuliuya V. Buyalskaya Belarusian State University, Niezaliežnasci Avenue, 4, 220030, Minsk, Belarus
  • Vasiliy M. Volkov Belarusian State University, Niezaliežnasci Avenue, 4, 220030, Minsk, Belarus

Keywords:

Chebyshev spectral methods, two-point boundary value problem, nonlinear interaction of counter-propagating optical waves, Newton’s method, conservative iterative method

Abstract

Chebyshev spectral methods for two-point boundary value problems describing the processes of counter interaction  of optical waves in media with cubic nonlinearity and linear media with periodic modulation of the refractive index are  considered. On the example of a linear problem, it is shown that the spectral method for achieving a given accuracy requires of two-three orders less time in comparison with the spline collocation method of the 5th accuracy order. Moreover,  Chebyshev mesh has natural adaptive properties for the considered problems of the nonlinear interaction of optical waves. A conservative iterative algorithm for implementation of the nonlinear spectral model is proposed. The proposed method  has a lower sensitivity to the choice of an appropriate initial guess and provides a higher rate of convergence in comparison with Newton’s method under conditions of strong coupling of interacting waves.

Author Biographies

  • Yuliuya V. Buyalskaya, Belarusian State University, Niezaliežnasci Avenue, 4, 220030, Minsk, Belarus

    senior lecturer at the department of  web-technologies and computer simulation, faculty of mechanics and mathematics

  • Vasiliy M. Volkov, Belarusian State University, Niezaliežnasci Avenue, 4, 220030, Minsk, Belarus

    doctor of science (physics and mathematics);  professor at the department of web-technologies and computer  simulation, faculty of mechanics and mathematics

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Published

2019-01-19

Issue

No. 3 (2018): Journal of the Belarusian State University. Mathematics and Informatics

Section

Computational Mathematics

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How to Cite

[1]
Buyalskaya, Y.V. and Volkov, V.M. 2019. Chebyshev spectral method for numerical simulations of counter-propagating optical waves interaction in nonlinear media. Journal of the Belarusian State University. Mathematics and Informatics. 3 (Jan. 2019), 75–81.