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Numerical Investigation of Spatial Solitons

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The numerical method we have previously developed to search for soliton solutions is applied in this article to find spatial solitons described by a nonlinear Schroedinger equation. We show in detail how to derive the Schroedinger equation from Maxwell’s equations and now to construct a one-soliton analytical solution for the one-dimensional Schroedinger equation. The proposed numerical method is appropriate to search for spatial solitons, which is corroborate by comparing the numerical results with the analytical solution.

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References

  1. Yu. S. Kivshar and G. P. Agraval, Optical Solitons: From Fibers to Photonic Crystals [Russian translation], Fizmatlit, Moscow (2005).

  2. M. Hercher, Opt. Soc. Amer., 54, 563 (1964).

    Google Scholar 

  3. P. L. Кеllеу, Phys. Rev. Lett., 15, 1005 (1965).

    Article  Google Scholar 

  4. J. E. Bjorkholm and A. Ashkin, Phys. Rev. Lett., 32, 129 (1974).

    Article  Google Scholar 

  5. A. Barthelemy, S. Maneuf, and C. Froehly, Opt. Commun., 55, 201 (1985).

    Article  Google Scholar 

  6. M. Shalaby and A. Barthelemy, Opt. Lett., 16, 1472 (1991).

    Article  Google Scholar 

  7. M. Shalaby, F. Reynaud, and A. Barthelemy, Opt. Lett., 17, 778 (1992).

    Article  Google Scholar 

  8. M. Shih, Z. Chen, M. Segev, et. al., Appl. Phys. Lett., 69, 4151 (1996).

  9. P. N. Butcher and D. N. Cotter, The Elements of Nonlinear Optics, Cambridge University Press, Cambridge (1990).

    Book  Google Scholar 

  10. V. S. Laponin, N. P. Savenkova, and V. P. Il'utko, “Numerical method for soliton solutions,” Comput. Math. Model., 23, No. 3, 254–265 (2012).

    Article  MathSciNet  MATH  Google Scholar 

  11. V. S. Laponin, “Search for soliton solutions in the three-dimensional Gross-Pitaevskii equation,” Comput. Math. Model., 25, No. 3, 306–314 (2014).

    Article  MathSciNet  MATH  Google Scholar 

  12. V. S. Laponin and N. P. Savenkova, “Search for 2-D solitons in Gross–Pitaevskii equation,” Comput. Math. Model., 25, No. 1, 1–8 (2014).

    Article  Google Scholar 

  13. N. P. Savenkova and V. S. Laponin, “A numerical method for finding soliton solutions in nonlinear differential equations,” Comput. Math. Cybern., 37, No. 2, 49–54 (2013).

    MathSciNet  MATH  Google Scholar 

  14. V. L. Bychkov, N. P. Savenkova, S. V. Anpilov, and Yu. V. Troshchiev, “Modeling of vorticle objects created in Gatchina discharge,” IEEE Transactions on Plasma Science, 40, No. 12, 3158–3161 (2012).

  15. Yusupaliev, N. P. Savenkova, Yu. V. Troshchiev, S. A. Shuteev, S. A. Skladchikov, E. E. Vinke, and N. G. Gusein-zade, “Vortex rings and plasma toroidal vortices in homogeneous unbounded media. II. The study of vortex formation process,” Bulletin of the Lebedev Physics Institute, 38, 275–282 (2011).

  16. U. Yusupaliev, N. P. Savenkova, S. A. Shuteev, S. A. Skladchikov, A. K. Maslov, V. G. Elensky, “Computer simulation of vortex self-maitenance and amplification,” Moscow University Physics Bulletin, 68, No. 4, 317–319 (2013).

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Authors and Affiliations

  1. Faculty of Computational Mathematics and Cybernetics, Moscow State University, Moscow, Russia

    V. S. Laponin

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  1. V. S. Laponin
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Correspondence to V. S. Laponin.

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Translated from Prikladnaya Matematika i Informatika, No. 49, 2015, pp. 45–54.

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Laponin, V.S. Numerical Investigation of Spatial Solitons. Comput Math Model 27, 181–189 (2016). https://doi.org/10.1007/s10598-016-9313-6

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  • DOI: https://doi.org/10.1007/s10598-016-9313-6

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