Optimized Algorithm for Petviashvili’s Method for Finding Solitons in Photonic Lattices

Part of the Springer Optimization and Its Applications book series (SOIA, volume 42)


In this paper we present an improved algorithm for Petviashvili’s heuristic numerical method for finding solitons in optically induced photonic lattices. Petviashvili’s method is usually used for approximating stationary solutions of nonlinear wave equations to construct numerically the solitary wave solutions such as solitons, lumps, and vortices. We developed and implemented a general software simulator designed for finding solitons in photonic lattices. The Petviashvili’s method is first modified by adding a stabilizing factor that greatly increases stability of the original method. Our software simulator implementation includes a number of criteria for recognizing, at an early stage, divergent or very slowly convergent cases. These criteria, that include Absence of stabilization, Slow convergence, Long interval of instability and Seesaw, significantly lower the overall calculation time.


Solitonic Solution Nonlinear Wave Equation Solitary Wave Solution Optical Soliton Photonic Lattice 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.



This research was supported by the Ministry for Science and Technical Development of the Republic of Serbia, Projects 144007 and 141031.


  1. 1.
    Banks, J. et al.: Discrete-event system simulation. 4th ed. Prentice-Hall, Upper Saddle River, N. J. (2005)Google Scholar
  2. 2.
    Chung, C.: Simulation Modeling Handbook. CRC Press, West Palm Beach, FL, USA, 608 (2003)Google Scholar
  3. 3.
    Fleisher, J.W. et al.: Observation of two-dimensional discrete solitons in optically induced nonlinear photonic lattices. Nature 422, 147–150 (2003)CrossRefGoogle Scholar
  4. 4.
    Kivshar, Y.S., Agrawal, G.P.: Optical Solitons. Academic Press, San Diego (2003)Google Scholar
  5. 5.
    Law, A.: How to build valid and credible simulation models. Proceedings of the 37th conference on Winter simulation, 24–32 (2005)Google Scholar
  6. 6.
    Pelinovsky, D., Stepanyants, Y.: Convergence of Petviashvili’s iteration method for numerical approximation of stationary solutions of nonlinear wave equations. SIAM J. Numer. Anal. 42(3), 1110–1127 (2004)MATHCrossRefMathSciNetGoogle Scholar
  7. 7.
    Petviashvili, V. I.: Equation for an extraordinary soliton. Sov. J. Plasma Phys. 2 257–258 (1976)Google Scholar
  8. 8.
    Seila, A. et al.: Applied Simulation Modeling, 494. Duxbury Press, North Scituate, MA, USA (2003)Google Scholar
  9. 9.
    Tian, B., Gao, Y.: Solutions of a variable-coefficient Kadomtsev-Petviashvili equation via computer algebra. Appl. Math. Comput. 84(2–3), 125–130 (1997)MATHCrossRefMathSciNetGoogle Scholar
  10. 10.
    Yang, J. et al.: Dipole and quadrupole solitons in optically induced two-dimensional photonic lattice: theory and experiment. Stud. Appl. Math. 113, 389 (2004)MATHCrossRefMathSciNetGoogle Scholar

Copyright information

© Springer Science+Business Media, LLC 2010

Authors and Affiliations

  1. 1.Institute of PhysicsPregrevica 118ZemunSerbia
  2. 2.Faculty of Computer ScienceMegatrend UniversityNovi BeogradSerbia

Personalised recommendations