Copyright © 2011 Sh. Amiranashvili and A. Demircan. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Linked References

1. M. Born and E. Wolf, Principles of Optics, Cambridge University Press, Cambridge, UK, 7th edition, 1999.
2. Y. Kodama and A. Hasegawa, “Nonlinear pulse propagation in a monomode dielectric guide,” IEEE Journal of Quantum Electronics, vol. 23, no. 5, pp. 510–524, 1987. View at Scopus
3. G. P. Agrawal, Nonlinear Fiber Optics, Academic Press, New York, NY, USA, 4th edition, 2007.
4. R. W. Boyd, Nonlinear Optics, Academic Press, New York, NY, USA, 3rd edition, 2008.
5. J. K. Ranka and A. L. Gaeta, “Breakdown of the slowly varying envelope approximation in the self-focusing of ultrashort pulses,” Optics Letters, vol. 23, no. 7, pp. 534–536, 1998. View at Scopus
6. J. E. Rothenberg, “Space-time focusing: breakdown of the slowly varying envelope approximation in the self-focusing of femtosecond pulses,” Optics Letters, vol. 17, no. 19, pp. 1340–1342, 1992.
7. F. DeMartini, C. H. Townes, T. K. Gustafson, and P. L. Kelley, “Self-steepening of light pulses,” Physical Review, vol. 164, no. 2, pp. 312–323, 1967. View at Publisher · View at Google Scholar · View at Scopus
8. P. Kinsler and G. H. C. New, “Few-cycle pulse propagation,” Physical Review A, vol. 67, no. 2, article 023813, 2003.
9. J. M. Dudley, G. Genty, and S. Coen, “Supercontinuum generation in photonic crystal fiber,” Reviews of Modern Physics, vol. 78, no. 4, pp. 1135–1183, 2006. View at Publisher · View at Google Scholar · View at Scopus
10. K. Akimoto, “Properties and applications of ultra-short electromagnetic mono- and sub-cycle waves,” Journal of the Physical Society of Japan, vol. 65, no. 7, pp. 2020–2032, 1996. View at Scopus
11. A. V. Kim, M. Y. Ryabikin, and A. M. Sergeev, “From femtosecond to attosecond pulses,” Uspekhi Fizicheskikh Nauk, vol. 42, no. 1, pp. 54–61, 1999.
12. T. Brabec and F. Krausz, “Intense few-cycle laser fields: frontiers of nonlinear optics,” Reviews of Modern Physics, vol. 72, no. 2, pp. 545–591, 2000. View at Scopus
13. P. M. Paul, E. S. Toma, P. Breger et al., “Observation of a train of attosecond pulses from high harmonic generation,” Science, vol. 292, no. 5522, pp. 1689–1692, 2001. View at Publisher · View at Google Scholar · View at PubMed · View at Scopus
14. M. Hentschel, R. Kienberger, C. Spielmann et al., “Attosecond metrology,” Nature, vol. 414, no. 6863, pp. 509–513, 2001. View at Publisher · View at Google Scholar · View at PubMed · View at Scopus
15. M. Drescher, M. Hentschel, R. Kienberger et al., “X-ray pulses approaching the attosecond frontier,” Science, vol. 291, no. 5510, pp. 1923–1927, 2001. View at Publisher · View at Google Scholar · View at PubMed · View at Scopus
16. S. A. Kozlov and S. V. Sazonov, “Nonlinear propagation of optical pulses of a few oscillations duration in dielectric media,” Journal of Experimental and Theoretical Physics, vol. 84, no. 2, pp. 221–228, 1997. View at Scopus
17. T. Schäfer and C. E. Wayne, “Propagation of ultra-short optical pulses in cubic nonlinear media,” Physica D, vol. 196, no. 1-2, pp. 90–105, 2004. View at Publisher · View at Google Scholar · View at MathSciNet · View at Scopus
18. V. G. Bespalov, S. A. Kozlov, Y. A. Shpolyanskiy, and I. A. Walmsley, “Simplified field wave equations for the nonlinear propagation of extremely short light pulses,” Physical Review A, vol. 66, no. 1, article 013811, 2002.
19. S. Amiranashvili, A. G. Vladimirov, and U. Bandelow, “A model equation for ultrashort optical pulses around the zero dispersion frequency,” European Physical Journal D, vol. 58, no. 2, pp. 219–226, 2010. View at Publisher · View at Google Scholar · View at Scopus
20. H. Leblond and D. Mihalache, “Few-optical-cycle solitons: modified Korteweg-de Vries sine-Gordon equation versus other non-slowly-varying-envelope-approximation models,” Physical Review A, vol. 79, no. 6, article 063835, 2009. View at Publisher · View at Google Scholar
21. A. I. Maimistov, “Some models of propagation of extremely short electromagnetic pulses in a nonlinear medium,” Quantum Electronics, vol. 30, no. 4, pp. 287–304, 2000. View at Scopus
22. K. E. Oughstun and H. Xiao, “Failure of the quasimonochromatic approximation for ultrashort pulse propagation in a dispersive, attenuative medium,” Physical Review Letters, vol. 78, no. 4, pp. 642–645, 1997. View at Scopus
23. S. Amiranashvili, U. Bandelow, and A. Mielke, “Padé approximant for refractive index and nonlocal envelope equations,” Optics Communications, vol. 283, no. 3, pp. 480–485, 2010. View at Publisher · View at Google Scholar · View at Scopus
24. T. Brabec and F. Krausz, “Nonlinear optical pulse propagation in the single-cycle regime,” Physical Review Letters, vol. 78, no. 17, pp. 3282–3285, 1997. View at Scopus
25. K. J. Blow and D. Wood, “Theoretical description of transient stimulated Raman scattering in optical fibers,” IEEE Journal of Quantum Electronics, vol. 25, no. 12, pp. 2665–2673, 1989. View at Publisher · View at Google Scholar · View at Scopus
26. N. Karasawa, S. Nakamura, N. Nakagawa et al., “Comparison between theory and experiment of nonlinear propagation for a-few-cycle and ultrabroadband optical pulses in a fused-silica fiber,” IEEE Journal of Quantum Electronics, vol. 37, no. 3, pp. 398–404, 2001. View at Publisher · View at Google Scholar · View at Scopus
27. M. A. Porras, “Propagation of single-cycle pulsed light beams in dispersive media,” Physical Review A, vol. 60, no. 6, pp. 5069–5073, 1999. View at Scopus
28. M. Geissler, G. Tempea, A. Scrinzi, M. Schnürer, F. Krausz, and T. Brabec, “Light propagation in field-ionizing media: extreme nonlinear optics,” Physical Review Letters, vol. 83, no. 15, pp. 2930–2933, 1999. View at Scopus
29. P. Kinsler and G. H. C. New, “Few-cycle soliton propagation,” Physical Review A, vol. 69, no. 1, article 013805, 2004.
30. X. Fu, L. Qian, S. Wen, and D. Fan, “Propagation of the ultrashort pulsed beam with ultrabroad bandwidth in the dispersive medium,” Physical Review A, vol. 68, no. 6, article 063818, 2003.
31. M. Kolesik, J. V. Moloney, and M. Mlejnek, “Unidirectional optical pulse propagation equation,” Physical Review Letters, vol. 89, no. 28, article 283902, 2002.
32. J. R. Gulley and W. M. Dennis, “Ultrashort-pulse propagation through free-carrier plasmas,” Physical Review A, vol. 81, no. 3, article 033818, 2010. View at Publisher · View at Google Scholar
33. J. Moses and W. W. Wise, “Controllable self-steepening of ultrashort pulses in quadratic nonlinear media,” Physical Review Letters, vol. 97, no. 7, article 073903, 2006. View at Publisher · View at Google Scholar
34. D. E. Vakman and L. A. Vainshtein, “Amplitude, phase, frequency—fundamental concepts of oscillation theory,” Uspekhi Fizicheskikh Nauk, vol. 20, no. 12, pp. 1002–1016, 1977.
35. A. V. Husakou and J. Herrmann, “Supercontinuum generation of higher-order solitons by fission in photonic crystal fibers,” Physical Review Letters, vol. 87, no. 20, article 203901, 2001.
36. A. Ferrando, M. Zacarés, P. F. de Córdoba, D. Binosi, and Á. Montera, “Forward-backward equations for nonlinear propagation in axially invariant optical systems,” Physical Review E, vol. 71, no. 1, article 016601, 2005. View at Publisher · View at Google Scholar
37. P. Kinsler, S. B. P. Radnor, and G. H. C. New, “Theory of directional pulse propagation,” Physical Review A, vol. 72, no. 6, article 063807, 2005. View at Publisher · View at Google Scholar
38. M. Kolesik and J. V. Moloney, “Nonlinear optical pulse propagation simulation: from Maxwell's to unidirectional equations,” Physical Review E, vol. 70, no. 3, article 036604, 2004. View at Publisher · View at Google Scholar
39. A. J. Lichtenberg and M. A. Lieberman, Regular and Chaotic Dynamics, vol. 38 of Applied Mathematical Sciences, Springer, New York, NY, USA, 2nd edition, 1982.
40. V. E. Zakharov, V. S. L'vov, and G. Falkovich, Kolmogorov Spectra of Turbulence 1. Wave Turbulence, Springer, Berlin, Germany, 1992.
41. S. Amiranashvili and A. Demircan, “Hamiltonian structure of propagation equations for ultrashort optical pulses,” Physical Review A, vol. 82, no. 1, article 013812, 2010. View at Publisher · View at Google Scholar
42. C. Canuto, M. Y. Hussaini, A. Quarteroni, and T. A. Zang, Spectral Methods: Fundamentals in Single Domains, Springer, Berlin, Germany, 3rd edition, 2006.
43. E. Hairer, S. P. Nørsett, and G. Wanner, Solving Ordinary Differential Equations I: Nonstiff Problems, Springer, Berlin, Germany, 2nd edition, 2000.
44. A. Demircan and U. Bandelow, “Supercontinuum generation by the modulation instability,” Optics Communications, vol. 244, no. 1, pp. 181–185, 2005. View at Publisher · View at Google Scholar · View at Scopus
45. A. Demircan and U. Bandelow, “Analysis of the interplay between soliton fission and modulation instability in supercontinuum generation,” Applied Physics B, vol. 86, no. 1, pp. 31–39, 2007. View at Publisher · View at Google Scholar · View at Scopus
46. I. Cristiani, R. Tediosi, L. Tartara, and V. Degiorgio, “Dispersive wave generation by solitons in microstructured optical fibers,” Optics Express, vol. 12, no. 1, pp. 124–135, 2004. View at Publisher · View at Google Scholar · View at Scopus
47. J. K. Ranka, R. S. Windeler, and A. J. Stentz, “Visible continuum generation in air-silica microstructure optical fibers with anomalous dispersion at 800 nm,” Optics Letters, vol. 25, no. 1, pp. 25–27, 2000. View at Scopus
48. V. E. Zakharov, S. L. Musher, and A. M. Rubenchik, “Hamiltonian approach to the description of non-linear plasma phenomena,” Physics Reports, vol. 129, no. 5, pp. 285–366, 1985. View at Scopus
49. V. E. Zakharov and E. A. Kuznetsov, “Hamiltonian formalism for nonlinear waves,” Uspekhi Fizicheskikh Nauk, vol. 40, no. 11, pp. 1087–1116, 1997. View at Scopus