Skip to main content
SpringerLink
Log in

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

  • Published:
Computational Mathematics and Mathematical Physics Aims and scope Submit manuscript

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.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

References

  1. A. V. Buryak and Yu. S. Kivshar, “Spatial Optical Solitons Governed by Quadratic Nonlinearity,” Opt. Lett. 19, 1612–1615 (1994).

    Article  Google Scholar 

  2. A. V. Buryak, P. D. Trapani, D. V. Skryabin, et al., “Optical Solitons Due to Quadratic Nonlinearities: From Basic Physics to Futuristic Applications,” Phys. Rep. 370(2), 63–235 (2002).

    Article  MATH  MathSciNet  Google Scholar 

  3. C. Etrich, F. Lederer, B. A. Malomed, et al., “Optical Solitons in Media with a Quadratic Nonlinearity,” Prog. Opt. 41, 483–568 (2000).

    Google Scholar 

  4. L. Brull and H. Lange, “Stationary, Oscillatory, and Solitary Wave Type Solution of Singular Nonlinear Schrödinger Equations,” Math. Methods Appl. Sci. 8, 559–575 (1986).

    Article  MathSciNet  Google Scholar 

  5. X. Liu, K. Beckwitt, and F. W. Wise, “Two-Dimensional Optical Spatiotemporal Solitons in Quadratic Media,” Phys. Rev. E 62, 1328–1340 (2000).

    Article  Google Scholar 

  6. G. Stegeman, D. J. Hagan, and L. Torner, “χ(2) Cascading Phenomena and Their Applications to All-Optical Signal Processing, Mode-Locking, Pulse Compression, and Solitons,” Opt. Quantum Electron. 28, 1691–1740 (1996).

    Article  Google Scholar 

  7. S. Ashihara, J. Nishina, T. Shimura, et al., “Soliton Compression of Femtosecond Pulses in Quadratic Media,” J. Opt. Soc. Am. 19, 2505–2510 (2002).

    Article  Google Scholar 

  8. I. Towers and B. A. Malomed, “Stable (2 + 1)-Dimensional Solitons in a Layered Medium with Sign-Alternating Kerr Nonlinearity,” J. Opt. Soc. Am. 19, 537–543 (2002).

    MathSciNet  Google Scholar 

  9. V. Steblina, Yu. S. Kivshar, M. Lisak, et al., “Self-Guided Beams in a Diffractive χ(2) Medium: Variational Approach,” Opt. Commun. 118, 345–352 (1995).

    Article  Google Scholar 

  10. J. Yang, B. A. Malomed, and D. J. Kaup, “Embedded Solitons in Second-Harmonic-Generating Systems,” Phys. Rev. Lett. 83, 1958–1961 (1999).

    Article  Google Scholar 

  11. D. Mihalache, D. Mazilu, B. A. Malomed, et al., “Stable Three-Dimensional Optical Solitons Supported by Competing Quadratic and Self-Focusing Cubic Nonlinearities,” Phys. Rev. E 74, 047601 (2006).

    Article  Google Scholar 

  12. B. A. Malomed, Soliton Management in Periodic Systems (Springer-Verlag, New York, 2006).

    Google Scholar 

  13. H. Sakaguchi and B. A. Malomed, “Resonant Nonlinearity Management for Nonlinear Schrödinger Solitons,” Phys. Rev. E 70, 066613 (2004).

    Article  Google Scholar 

  14. L. Berge, V. K. Mezentsev, J. J. Rasmussen, et al., “Self-Guiding Light in Layered Nonlinear Media,” Opt. Lett. 25, 1037–1039 (2000).

    Article  Google Scholar 

  15. N. N. Rozanov, S. V. Fedorov, and A. N. Shatsev, “Incoherent Weak Coupling of Laser Solitons,” Opt. Spectrosc. 102(1), 83–85 (2007).

    Article  Google Scholar 

  16. I. V. Babushkin, N. A. Loiko, and N. N. Rozanov, “Spatial Soliton-Like Structures in a Thin-Film System with a Transverse Photonic Crystal in the Feedback Loop,” Opt. Spektrosk. 102, 285–291 (2007) [Opt. Spectrosc. 102, 248–254 (2007)].

    Article  Google Scholar 

  17. R. Schiek, Y. Baek, G. Stegeman, et al., “Interactions between One-Dimensional Quadratic Soliton-Like Beams,” Opt. Quantum Electron. 30, 861–879 (1998).

    Article  Google Scholar 

  18. R. Driben, Y. Oz, B. A. Malomed, et al., “Mismatch Management for Optical and Matter-Wave Quadratic Solitons,” Phys. Rev. E 75, 026612 (2007).

    Article  Google Scholar 

  19. H. Saito and M. Ueda, “Dynamically Stabilized Bright Solitons in a Two-Dimensional Bose-Einstein Condensate,” Phys. Rev. Lett. 90, 040403 (2003).

    Article  Google Scholar 

  20. F. K. Abdullaev, J. G. Caputo, R. A. Kraenkel, and B. A. Malomed, “Controlling Collapse in Bose-Einstein Condensates by Temporal Modulation of the Scattering Length,” Phys. Rev. A 67, 013605 (2003).

    Article  Google Scholar 

  21. I. Towers, A. V. Buryak, R. A. Sammut, and B. A. Malomed, “Stable Localized Vortex Solitons,” Phys. Rev. E 63, 055601 (2001).

    Article  Google Scholar 

  22. V. E. Zakharov, S. V. Manakov, S. P. Novikov, and L. P. Pitaevskii, Theory of Solitons: The Inverse Scattering Method (Nauka, Moscow, 1980; Consultants Bureau, New York, 1984).

    MATH  Google Scholar 

  23. B. S. Kerner and V. V. Osipov, Autosolitons: A New Approach to Problems of Self-Organization and Turbulence (Nauka, Moscow, 1991; Kluwer, Dordrecht, 1994).

    Google Scholar 

  24. F. Abdullaev, S. Darmanyan, and P. Khabibullaev, Optical Solitons (Berlin, Heidelberg, 1993).

    Google Scholar 

  25. M. J. Ablowitz and H. Segur, Solitons and the Inverse Scattering Transform (SIAM, Philadelphia, Pa., 1981; Mir, Moscow, 1987).

    MATH  Google Scholar 

  26. M. J. Ablowitz and P. A. Clarkson, Solitons: Nonlinear Evolution Equations and Inverse Scattering (Cambridge Univ. Press., Cambridge, 1991).

    MATH  Google Scholar 

  27. L. D. Faddeev and L. A. Takhtajan, Hamiltonian Methods in the Theory of Solitons (Nauka, Moscow, 1986; Springer-Verlag, Berlin, 1987).

    MATH  Google Scholar 

  28. G. L. Lamb, Jr., Elements of Soliton Theory (Wiley, New York, 1980; Mir, Moscow, 1983).

    MATH  Google Scholar 

  29. O. I. Bogoyavlenskii, Breaking Solitons: Nonlinear Integrable Equations (Nauka, Moscow, 1991) [in Russian].

    Google Scholar 

  30. V. I. Nayanov, Multipole Solitons (Fizmatlit, Moscow, 2006) [in Russian].

    Google Scholar 

  31. R. K. Dodd, J. C. Eilbeck, J. Gibbon, and H. C. Morris, Solitons and Nonlinear Wave Equations (Academic, New York, 1982; Mir, Moscow, 1988).

    MATH  Google Scholar 

  32. T. Miwa, M. Jimbo, and E. Date, Solitons: Differential Equations, Symmetries, and Infinite-Dimensional Algebras (Cambridge Univ. Press, Cambridge, 2000; MTsNMO, Moscow, 2005).

    MATH  Google Scholar 

  33. E. Infeld and G. Rowlands, Nonlinear Waves, Solitons, and Chaos (Cambridge Univ. Press, Cambridge, 2000; Fizmatlit, Moscow, 2005).

    MATH  Google Scholar 

  34. A. C. Newell, Solitons in Mathematics and Physics (SIAM, Philadelphia, Pa., 1985; Mir, Moscow, 1989).

    Google Scholar 

  35. F. Calogero and A. Degasperis, Spectral Transform and Solitons: Tools to Solve and Investigate Nonlinear Evolution Equations (North-Holland, Amsterdam, 1982; Mir, Moscow, 1985).

    MATH  Google Scholar 

  36. A. S. Davydov, Solitons in Molecular Systems (Naukova Dumka, Kiev, 1984; Kluwer, Dordrecht, 1991).

    Google Scholar 

  37. V. Yu. Novokshenov, Introduction to Soliton Theory (RKhD, Moscow, 2002) [in Russian].

    Google Scholar 

  38. R. Rajaraman, Solitons and Instantons: An Introduction to Solitons and Instantons in Quantum Field Theory (North-Holland, Amsterdam, 1982; Mir, Moscow, 1985).

    MATH  Google Scholar 

  39. Y. S. Kivshar and G. P. Agrawal, Optical Solitons: From Fibers to Photonic Crystals (Academic, New York, 2003; Fizmatlit, Moscow, 2005).

    Google Scholar 

  40. N. N. Akhmediev and A. Ankiewicz, Solitons: Nonlinear Pulses and Beams (Kluwer Academic, London, 1997; Fizmatlit, Moscow, 2003).

    Google Scholar 

  41. Yu. N. Karamzin and A. P. Sukhorukov, “Nonlinear Interaction of Diffracted Light Beams in a Medium with Quadratic Nonlinearity: Mutual Focusing of Beams and Limitation on the Efficiency of Optical Frequency Converters,” Pis’ma Zh. Eksp. Teor. Fiz. 20, 734–739 (1974).

    Google Scholar 

  42. S. A. Varentsova and V. A. Trofimov, “Finite-Difference Method for Finding the Eigenmodes of the Nonlinear Schrödinger Equation,” Vestn. Mosk. Gos. Univ., Ser. 15., No. 3, 16–22 (2005).

  43. V. A. Trofimov and S. A. Varentsova, “Computational Method for Finding of Soliton Solutions of a Nonlinear Schrödinger Equation,” Lecture Notes in Computer Science (Springer-Verlag, Berlin, 2005), Vol. 3401, pp. 550–557.

    Google Scholar 

  44. O. V. Matusevich and V. A. Trofimov, “Iterative Method for Finding the Eigenfunctions of a System of Two Schrödinger Equations with Combined Nonlinearity,” Zh. Vychisl. Mat. Mat. Fiz. 48, 713–724 (2008) [Comput. Math. Math. Phys. 48, 677–687 (2008)].

    MATH  MathSciNet  Google Scholar 

  45. A. A. Samarskii and V. B. Andreev, Difference Methods for Elliptic Equations (Nauka, Moscow, 1976) [in Russian].

    Google Scholar 

  46. G. H. Golub and C. F. van Loan, Matrix Computations (Johns Hopkins Univ. Press, Baltimore, Md., 1996; Mir, Moscow, 1999).

    MATH  Google Scholar 

  47. A. A. Samarskii and A. V. Gulin, Numerical Methods (Nauka, Moscow, 1989) [in Russian].

    MATH  Google Scholar 

  48. http://parallel.ru/cluster/leo-linpack.html

  49. J. W. Demmel, Applied Numerical Linear Algebra (SIAM, Philadelphia, PA, 1997; Mir, Moscow, 2001).

    MATH  Google Scholar 

  50. http://www.caam.rice.edu/software/ARPACK/

  51. B. A. Malomed, D. Mihalache, F. Wise, and L. Torner, “Spatiotemporal Optical Solitons,” J. Opt. B 7(5), 53–72 (2005).

    Google Scholar 

  52. J. H. B. Nijhof, N. J. Doran, W. Forysiak, and F. M. Knox, “Stable Soliton-Like Propagation in Dispersion Managed Systems with Net Anomalous, Zero, and Normal Dispersion,” Electron. Lett. 33, 1726–1727 (1997).

    Article  Google Scholar 

  53. T. Lakoba, J. Yang, D. J. Kaup, and B. A. Malomed, “Conditions for Stationary Pulse Propagation in the Strong Dispersion Management Regime,” Opt. Commun. 149, 366–375 (1998).

    Article  Google Scholar 

  54. K. D. Moll, A. L. Gaeta, and G. Fibich, “Self-Similar Optical Wave Collapse: Observation of the Townes Profile,” Phys. Rev. Lett. 90, 203902 (2003).

    Article  Google Scholar 

  55. G. D. Montesinos and V. M. Perez-Garcia, “Numerical Studies of Stabilized Townes Solitons,” Math. Comput. Simulation 69, 447–456 (2005).

    Article  MATH  MathSciNet  Google Scholar 

  56. G. D. Montesinos, V. M. Perez-Garcia, and P. J. Torres, “Stabilization of Solitons of the Multidimensional Nonlinear Schrödinger Equation: Matter-Wave Breathers,” Phys. D 191, 193–210 (2004).

    Article  MATH  MathSciNet  Google Scholar 

Download references

Author information

Authors and Affiliations

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

    O. V. Matusevich & V. A. Trofimov

Authors
  1. O. V. Matusevich
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. V. A. Trofimov
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to V. A. Trofimov.

Additional information

Original Russian Text © O.V. Matusevich, V.A. Trofimov, 2009, published in Zhurnal Vychislitel’noi Matematiki i Matematicheskoi Fiziki, 2009, Vol. 49, No. 11, pp. 1988–2000.

Rights and permissions

Reprints and permissions

About this article

Cite this article

Matusevich, O.V., Trofimov, V.A. Numerical method for finding 3D solitons of the nonlinear Schrödinger equation in the axially symmetric case. Comput. Math. and Math. Phys. 49, 1902–1912 (2009). https://doi.org/10.1134/S0965542509110074

Download citation

  • Received:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1134/S0965542509110074

Key words

Search

Navigation