Skip to main content
SpringerLink
Log in

Nonlinear Schrödinger’s Equation

  • Chapter
Mathematical Theory of Dispersion-Managed Optical Solitons

Part of the book series: Nonlinear Physical Science ((NPS))

  • 893 Accesses

Abstract

In this chapter the nonlinear Schrodinger’s equation (NLSE) will be derived from the basic principles of Electromagnetic Theory. This equation will be then modified in presence of dispersion-management. The conserved quantities of this dispersion-managed NLSE (DM-NLSE) will be derived. The variational principle used for solving the DM-NLSE will be introduced. Finally, this chapter will end with a brief introduction to the soliton perturbation theory.

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

Access this chapter

Log in via an institution
Chapter
USD 29.95
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
eBook
USD 84.99
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
Hardcover Book
USD 109.99
Price excludes VAT (USA)
  • Durable hardcover edition
  • Dispatched in 3 to 5 business days
  • Free shipping worldwide - see info

Tax calculation will be finalised at checkout

Purchases are for personal use only

Institutional subscriptions

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. F. Abdullaev, S. Darmanyan & P. Khabibullaev. Optical Solitons. Springer, New York, NY. USA. (1993).

    Google Scholar 

  2. M. J. Ablowitz & H. Segur. Solitons and the Inverse Scattering Transform. SIAM. Philadelphia, PA. USA. (1981).

    MATH  Google Scholar 

  3. M. J. Ablowitz, G. Biondini, S. Chakravarti, R. B. Jenkins & J. R. Sauer. “Four-wave mixing in wavelength-division-multiplexed soliton systems: damping and amplification”. Optics Letters. Vol 21, No 20, 1646–1648. (1996).

    Article  ADS  Google Scholar 

  4. M. J. Ablowitz, B. M. Herbst & C. M. Schober. “The nonlinear Schrödinger’s equation; Asymmetric perturbations traveling waves and chaotic structures”. Mathematics and Computers in Simulation. Vol 43, 3–12. (1997).

    Article  MathSciNet  MATH  Google Scholar 

  5. M. J. Ablowitz, G. Biondini, S. Chakravarty, R. B. Jenkins & J. R. Sauer. “Four-wave mixing in wavelength-division multiplexed soliton systems—Ideal fibers”. Journal of Optical Society of America B. Vol 14, 1788–1794. (1997).

    Article  ADS  Google Scholar 

  6. M. J. Ablowitz & G. Biondini. “Multiscale pulse dynamics in communication systems with strong dispersion management”. Optics Letters. Vol 23, No 21, 1668–1670. (1998).

    Article  ADS  Google Scholar 

  7. M. J. Ablowitz & P. A. Clarkson. “Solitons and symmetries”. Journal of Engineering Mathematics. Vol 36, 1–9. (1999).

    Article  MathSciNet  MATH  Google Scholar 

  8. M. J. Ablowitz & T. Hirooka. “Resonant nonlinear interactions in strongly dispersion-managed transmission systems”. Optics Letters. Vol 25, Issue 24, 1750–1752. (2000).

    Article  ADS  Google Scholar 

  9. M. J. Ablowitz, G. Biondini & L. A. Ostrovsky. “Optical solitons: Perspectives and applications”. Chaos. Vol 10, No 3, 471–474. (2000).

    Article  MathSciNet  ADS  Google Scholar 

  10. M. J. Ablowitz & T. Hirooka. “Nonlinear effects in quasi-linear dispersion-managed systems”. IEEE Photonics Technology Letters. Vol 13, No 10, 1082–1084. (2001).

    Article  ADS  Google Scholar 

  11. M. J. Ablowitz, T. Hirooka & G. Biondini. “Quasi-linear optical pulses in strongly dispersion-managed transmission system”. Optics Letters. Vol 26, No 7, 459–461. (2001).

    Article  ADS  Google Scholar 

  12. M. J. Ablowitz, G. Biondini & E. S. Olson. “Incomplete collisions of wavelength-division multiplexed dispersion-managed solitons”. Journal of Optical Society of America B. Vol 18, No 3, 577–583. (2001).

    Article  MathSciNet  ADS  Google Scholar 

  13. M. J. Ablowitz & T. Hirooka. “Nonlinear effects in quasilinear dispersion-managed pulse transmission”. IEEE Journal of Photonics Technology Letters. Vol 26, 1846–1848. (2001).

    Google Scholar 

  14. M. J. Ablowitz & T. Hirooka. “Intrachannel pulse interactions and timing shifts in strongly dispersion-managed transmission systems”. Optics Letters. Vol 26, No 23, 1846–1848. (2001).

    Article  ADS  Google Scholar 

  15. M. J. Ablowitz & T. Hirooka. “Intrachannel pulse interactions in dispersion-managed transmission systems: energy transfer”. Optics Letters. Vol 27, No 3, 203–205. (2002).

    Article  ADS  Google Scholar 

  16. M. J. Ablowitz & T. Hirooka. “Managing nonlinearity in strongly dispersion-managed optical pulse transmission”. Journal of Optical Society of America B. Vol 19, No 3, 425–439. (2002).

    Article  ADS  Google Scholar 

  17. M. J. Ablowitz, G. Biondini, A. Biswas, A. Docherty, T. Hirooka & S. Chakravarty. “Collision-induced timing shifts in dispersion-managed soliton systems”. Optics Letters. Vol 27, 318–320. (2002).

    Article  ADS  Google Scholar 

  18. M. J. Ablowitz, T. Hirooka & T. Inoue. “Higher-order asymptotic analysis of dispersion-managed transmission systems: solutions and their characteristics”. Journal of Optical Society of America B. Vol 19, No 12, 2876–2885. (2002).

    Article  ADS  Google Scholar 

  19. G. P. Agrawal. Nonlinear Fiber Optics. Academic Press, San Deigo, CA. USA. (1995).

    Google Scholar 

  20. C. D. Ahrens, M. J. Ablowitz, A. Docherty, V. Oleg, V. Sinkin, V. Gregorian & C. R. Menyuk. “Asymptotic analysis of collision-induced timing shifts in return-to-zero quasi-linear systems with pre-and post-dispersion compensation”. Optics Letters. Vol 30, 2056–2058. (2005).

    Article  ADS  Google Scholar 

  21. N. N. Akhmediev & A. Ankiewicz. Solitons, Nonlinear Pulses and Beams. Chapman and Hall. (1997).

    Google Scholar 

  22. D. Anderson. “Variational approach to nonlinear pulse propagation in optical fibers”. Physical Review A. Vol 27, No 6, 3135–3145. (1983).

    Article  ADS  Google Scholar 

  23. A. Biswas. “Dynamics of Gaussian and super-Gaussian solitons in birefringent optical fibers”. Progress in Electromagnetics Research. Vol 33, 119–139. (2001).

    Article  Google Scholar 

  24. A. Biswas. “Dispersion-Managed vector solitons in optical fibers”. Fiber and Integrated Optics. Vol 20, No 5, 503–515. (2001).

    Article  ADS  Google Scholar 

  25. A. Biswas. “Gaussian solitons in birefringent optical fibers”. International Journal of Pure and Applied Mathematics. Vol 2, No 1, 87–104. (2002).

    MathSciNet  MATH  Google Scholar 

  26. A. Biswas. “Super-Gaussian solitons in optical fibers”. Fiber and Integrated Optics. Vol 21, No 2, 115–124. (2002).

    Article  ADS  Google Scholar 

  27. A. Biswas. “Dispersion-managed solitons in optical fibers” Journal of Optics A. Vol 4, No 1, 84–97. (2002).

    Google Scholar 

  28. A. Biswas. “Integro-differential perturbation of dispersion-managed solitons”. Journal of Electromagnetic Waves and Applications. Vol 17, No 4, 641–665. (2003).

    Article  Google Scholar 

  29. A. Biswas. “Dispersion-managed solitons in optical couplers”. Journal of Nonlinear-Optical Physics and Materials. Vol 12, No 1, 45–74. (2003).

    Article  ADS  Google Scholar 

  30. A. Biswas. “Gabitov-Turitsyn equation for solitons in multiple channels”. Journal of Electromagnetic Waves and Applications. Vol 17, No 11, 1539–1560. (2003).

    Article  MathSciNet  Google Scholar 

  31. A. Biswas. “Gabitov-Turitsyn equation for solitons in optical fibers”. Journal of Nonlinear Optical Physics and Materials. Vol 12, No 1, 17–37. (2003).

    Article  ADS  Google Scholar 

  32. A. Biswas. “Dispersion-Managed solitons in multiple channels”. Journal of Nonlinear Optical Physics and Materials. Vol 13, No 1, 81–102. (2004).

    Article  ADS  Google Scholar 

  33. A. Biswas. “Theory of quasi-linear pulses in optical fibers”. Optical Fiber Technology. Vol 10, No 3, 232–259. (2004).

    Article  ADS  Google Scholar 

  34. A. Biswas. “Perturbations of dispersion-managed optical solitons”. Progress in Electromagnetics Research. Vol 48, 85–123. (2004).

    Article  ADS  Google Scholar 

  35. A. Biswas. “Dispersion-Managed solitons in birefringent fibers and multiple channels”. International Mathematical Journal. Vol 5, No 5, 477–515. (2004).

    MathSciNet  MATH  Google Scholar 

  36. A. Biswas. “Stochastic perturbation of dispersion-managed optical solitons”. Optical and Quantum Electronics. Vol 37, No 7, 649–659. (2005).

    Article  Google Scholar 

  37. A. Biswas & S. Konar. Introduction to Non-Kerr Law Optical Solitons. CRC Press, Boca Raton, FL. USA. (2006).

    Book  Google Scholar 

  38. D. Breuer, F. Kuppers, A. Mattheus, E. G. Shapiro, I. Gabitov & S. K. Turitsyn. “Symmetrical dispersion compensation for standard monomode fiber based communication systems with large amplifier spacing”. Optics Letters. Vol 22, No 13, 982–984. (1997).

    Article  ADS  Google Scholar 

  39. D. Breuer, K. Jürgensen, F. Küppers, A. Mattheus, I. Gabitov & S. K. Turitsyn. “Optimal schemes for dispersion compensation of standard monomode fiber based links”. Optics Communications. Vol 40, No 1–3, 15–18. (1997).

    Article  ADS  Google Scholar 

  40. V. Cautaerts, A. Maruta & Y. Kodama. “On the dispersion-managed soliton”. Chaos. Vol 10, No 3, 515–528.

    Google Scholar 

  41. F. J. Diaz-Otero & P. Chamorro-Posada. “Interchannel soliton collisions in periodic dispersion maps in the presence of third order dispersion”. Journal of Nonlinear Mathematical Physics. Vol 15,Supplement 3, 137–143. (2008).

    Article  MathSciNet  ADS  Google Scholar 

  42. S. L. Doty, J. W. Haus, Y. J. Oh & R. L. Fork. “Soliton interaction on dual core fibers”. Physical Review E. Vol 51, No 1, 709–717. (1995).

    Article  ADS  Google Scholar 

  43. A. Ehrhardt, M. Eiselt, G. Goβkopf, L. Küller, W. Pieper, R. Schnabel, G. H. Weber. “Semiconductor laser amplifier as optical switching gate”. Journal of Lightwave Technology. Vol 11, No 8, 1287–1295. (1993).

    Article  ADS  Google Scholar 

  44. A. Hasegawa. “A historical review of applications of optical solitons for high speed communications”. Chaos. Vol 10, No 3, 475–485. (2000).

    Article  MathSciNet  ADS  MATH  Google Scholar 

  45. A-H. Guan & Y-H. Wang. “Experimental study of interband and intraband crosstalk in WDM networks”. Optoelectronics Letters. Vol 4, No 1, 42–44. (2008).

    Article  MathSciNet  ADS  Google Scholar 

  46. E. Iannone, R. Sabella, M. Avattaneo & G. De Paolis. “Modeling of in-band crosstalk in WDM optical networks”. Journal of lightwave technology. Vol 17, No 7, 1135–1141. (1999).

    Article  ADS  Google Scholar 

  47. W. L. Kath & N. F. Smyth. “Soliton evolution and radiation loss for the nonlinear Schrodinger’s equation”. Physical Review E. Vol 51, No 2, 1484–1492. (1995).

    Article  MathSciNet  ADS  Google Scholar 

  48. C. M. Khalique & A. Biswas. “A Lie Symmetry approach to nonlinear Schrödinger’s equation with non-Kerr law nonlinearity”. Communications in Nonlinear Science and Numerical Simulation. Vol 14, No 12, 4033–4040. (2009).

    Article  MathSciNet  ADS  Google Scholar 

  49. Y. S. Kivshar & B. A. Malomed. “Dynamics of solitons in nearly integrable systems”. Rev. Mod. Physics. Vol 61, No 4, 763–915. (1989).

    Article  ADS  Google Scholar 

  50. Y. Kodama & M. J. Ablowitz. “Perturbations of solitons and solitary waves”. Studies in Applied Mathematics. Vol 64, 225–245. (1981).

    MathSciNet  ADS  MATH  Google Scholar 

  51. R. Kohl, A. Biswas, D. Milovic & E. Zerrad. “Perturbation of Gaussian optical solitons in dispersion-managed fibers”. Applied Mathematics and Computation. Vol 199, No 1, 250–258. (2009).

    Article  MathSciNet  Google Scholar 

  52. R. Kohl, A. Biswas, D. Milovic & E. Zerrad. “Perturbation of super-sech solitons in dispersion-managed optical fibers”. International Journal of Theoretical Physics. Vol 47, No 7, 2038–2064. (2008).

    Article  ADS  MATH  Google Scholar 

  53. R. Kohl, A. Biswas, D. Milovic & E. Zerrad. “Adiabatic dynamics of Gaussian and super-Gaussian solitons in dispersion-managed optical fibers”. Progress in Electromagnetics Research. Vol 84, 27–53. (2008).

    Article  Google Scholar 

  54. R. Kohl, D. Milovic, E. Zerrad & A. Biswas. “Perturbation of super-Gaussian optical solitons in dispersion-managed fibers”. Mathematical and Computer Modelling. Vol 49, No 7–8, 418–427. (2009).

    MathSciNet  Google Scholar 

  55. R. Kohl, D. Milovic, E. Zerrad & A. Biswas. “Optical solitons by He’s variational principle in a non-Kerr law media”. Journal of Infrared, Millimeter and Terahertz Waves. Vol 30, No 5, 526–537. (2009).

    Article  Google Scholar 

  56. R. Kohl, D. Milovic, E. Zerrad & A. Biswas. “Soliton perturbation theory for dispersion-managed optical fibers”. Journal of Nonlinear Optical Physics and Materials. Vol 18, No 2. (2009).

    Google Scholar 

  57. Z. M. Liao, C. J. McKinstrie & G. P. Agrawal. “Importance of pre-chirping in constant-dispersion fiber links with a large amplifier spacing”. Journal of Optical Society of America B. Vol 17, No 4, 514–518. (2000).

    Article  ADS  Google Scholar 

  58. P. M. Lushnikov. “Dispersion-managed solitons in optical fibers with zero average dispersion”. Optics Letters. Vol 25, No 16, 1144–1146. (2000).

    Article  ADS  Google Scholar 

  59. M. F. Mahmood & S. B. Qadri. “Modeling propagation of chirped solitons in an elliptically low birefringent single-mode optical fiber”. Journal of Nonlinear Optical Physics and Materials. Vol 8, No 4, 469–475. (1999).

    Article  ADS  Google Scholar 

  60. M. Matsumoto. “Analysis of interaction between stretched pulses propagating in dispersion-managed fibers”. IEEE Photonics Technology Letters. Vol 10, No 3, 373–375. (1998).

    Article  ADS  Google Scholar 

  61. C. R. Menyuk. “Application of multiple-length-scale methods to the study of optical fiber transmission”. Journal of Engineering Mathematics. Vol 36, No 1–2, 113–136. (1999).

    Article  MathSciNet  MATH  Google Scholar 

  62. H. Michinel. “Pulses nonlinear surface waves and soliton emission at nonlinear graded index waveguides”. Optical and Quantum Electronics. Vol 30, No 2, 79–97. (1998).

    Article  Google Scholar 

  63. M. Nakazawa H. Kubota, K. Suzuki & E. Yamada. “Recent progress in soliton transmission technology”. Chaos. Vol 10, No 3, 486–514. (2000).

    Article  ADS  Google Scholar 

  64. T. Okamawari, A. Maruta & Y. Kodama. “Analysis of Gordon-Haus jitter in a dispersion-compensated optical transmission system”. Optics Letters. Vol 23, No 9, 694–696. (1998).

    Article  ADS  Google Scholar 

  65. T. Okamawari, A. Maruta & Y. Kodama. “Reduction of Gordon-Haus jitter in a dispersion-compensated optical transmission system: analysis”. Optics Communications. Vol 141, No 9, 262–266. (1998).

    Google Scholar 

  66. A. Panajotovic, D. Milovic & A. Mittic. “Boundary case of pulse propagation analytic solution in the presence of interference and higher order dispersion”. TELSIKS 2005 Conference Proceedings. 547–550. Nis-Serbia. (2005).

    Google Scholar 

  67. A. Panajotovic, D. Milovic, A. Biswas & E. Zerrad. “Influence of even order dispersion on super-sech soliton transmission quality under coherent crosstalk”. Research Letters in Optics. Vol 2008, 613986, 5 pages. (2008).

    Google Scholar 

  68. A. Panajotovic, D. Milovic & A. Biswas. “Influence of even order dispersion on soliton transmission quality with coherent interference”. Progress in Electromagnetics Research B. Vol 3, 63–72. (2008).

    Article  Google Scholar 

  69. Y. Pointurier, M. Brandt-Pearce & C. L. Brown. “Analytical study of crosstalk propagation in all-optical networks using perturbation theory”. Journal of Lightwave Technology. Vol 23, No 12, 4074–4083. (2005).

    Article  ADS  Google Scholar 

  70. J. M. Senior. Optic Fiber Communications. Prentice Hall, New York, NY. USA. (1992).

    Google Scholar 

  71. V. N. Serkin & A. Hasegawa. “Soliton Management in the Nonlinear Schrodinger Equation Model with Varying Dispersion, Nonlinearity and Gain”. JETP Letters. Vol 72, No 2, 89–92. (2000).

    Article  ADS  Google Scholar 

  72. O. V. Sinkin, V. S. Grigoryan & C. R. Menyuk. “Accurate Probabilistic treatment of bit-pattern-dependent Nonlinear Distortions in BER calculations for WDM RZ systems”. Journal of Lightwave Technology. Vol 25, No 10, 2959–2967. (2007).

    Article  ADS  Google Scholar 

  73. C. D. Stacey, R. M. Jenkins, J. Banerji & A. R. Davis. “Demonstration of fundamental mode only propagation in highly multimode fibre for high power EDFAs”. Optics Communications. Vol 269, 310–314. (2007).

    Article  ADS  Google Scholar 

  74. M. Stefanovic & D. Milovic. “The impact of out-of-band crosstalk on optical communication link preferences”. Journal of Optical Communications. Vol 26, No 2, 69–72. (2005).

    Google Scholar 

  75. M. Stefanovic, D. Draca, A. Panajotovic & D. Milovic. “Individual and joint influence of second and third order dispersion on transmission quality in the presence of coherent interference”. Optik. Vol 120, No 13, 636–641.(2009).

    ADS  Google Scholar 

  76. B. Stojanovic, D. M. Milovic & A. Biswas. “Timing shift of optical pulses due to interchannel crosstalk”. Progress in Electromagnetics Research M. Vol 1, 21–30. (2008).

    Article  Google Scholar 

  77. S. K. Turitsyn, I. Gabitov, E. W. Laedke, V. K. Mezentsev, S. L. Musher, E. G. Shapiro, T. Schäfer, & K. H. Spatschek. “Variational approach to optical pulse propagation in dispersion compensated transmission system”. Optics Communications. Vol 151, No 1–3, 117–135. (1998).

    Article  ADS  Google Scholar 

  78. S. Wabnitz, Y. Kodama. & A. B. Aceves “Control of optical soliton interactions”. Optical Fiber Technology. Vol 1, 187–217. (1995).

    Article  ADS  Google Scholar 

  79. P. K. A. Wai & C. R. Menyuk. “Polarization mode dispersion, de-correlation, and diffusion in optical fibers with randomly varying birefringence”. Journal of Lightwave Technology. Vol 14, No 2, 148–157. (1996).

    Article  ADS  Google Scholar 

  80. V. E. Zakharov & S. Wabnitz. Optical Solitons: Theoretical Challenges and Industrial Perspectives. Springer, New York, CA. USA. (1999).

    Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Dept of Applied Mathematics & Theoretical Physics, Delaware State University, 1200 N Dupont Highway, Dover, DE, 19901-2277, USA

    Anjan Biswas

  2. Faculty of Electronic Engineering Department of Telecommunications, University of Nis, Serbia

    Daniela Milovic

  3. School of Arts and Sciences Department of Physics, Alabama A & M University, Normal, AL-35762, USA

    Matthew Edwards

Authors
  1. Anjan Biswas
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Daniela Milovic
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. Matthew Edwards
    View author publications

    You can also search for this author in PubMed Google Scholar

Rights and permissions

Reprints and permissions

Copyright information

© 2010 Higher Education Press, Beijing and Springer-Verlag Berlin Heidelberg

About this chapter

Cite this chapter

Biswas, A., Milovic, D., Edwards, M. (2010). Nonlinear Schrödinger’s Equation. In: Mathematical Theory of Dispersion-Managed Optical Solitons. Nonlinear Physical Science. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-10220-2_2

Download citation

Publish with us

Policies and ethics

Search

Navigation