Copyright © 2009 Brandon G. Bale et al. 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. V. L. Ginzburg and L. D. Landau, “To the theory of superconductivity,” Soviet Physics Journal of Experimental and Theoretical Physics, vol. 20, pp. 1064–1082, 1950.
2. I. S. Aranson and L. Kramer, “The world of the complex Ginzburg-Landau equation,” Reviews of Modern Physics, vol. 74, no. 1, pp. 99–143, 2002. View at Publisher · View at Google Scholar · View at MathSciNet
3. H. A. Haus, “Mode-locking of lasers,” IEEE Journal on Selected Topics in Quantum Electronics, vol. 6, no. 6, pp. 1173–1185, 2000. View at Publisher · View at Google Scholar
4. H. A. Haus, J. G. Fujimoto, and E. P. Ippen, “Analytic theory of additive pulse and Kerr lens mode locking,” IEEE Journal of Quantum Electronics, vol. 28, no. 10, pp. 2086–2096, 1992. View at Publisher · View at Google Scholar
5. J. N. Kutz, “Mode-locked soliton lasers,” SIAM Review, vol. 48, no. 4, pp. 629–678, 2006. View at Publisher · View at Google Scholar · View at MathSciNet
6. J. Lega, J. V. Moloney, and A. C. Newell, “Swift-Hohenberg equation for lasers,” Physical Review Letters, vol. 73, no. 22, pp. 2978–2981, 1994. View at Publisher · View at Google Scholar
7. G.-L. Oppo, G. D'Alessandro, and W. J. Firth, “Spatiotemporal instabilities of lasers in models reduced via center manifold techniques,” Physical Review A, vol. 44, no. 7, pp. 4712–4720, 1991. View at Publisher · View at Google Scholar
8. N. N. Akhmediev and A. Ankiewicz, Eds., Solitons, Nonlinear Pulses and Beams, Chapman and Hall, London, UK, 1997.
9. J. M. Soto-Crespo, N. N. Akhmediev, and V. V. Afanasjev, “Stability of the pulselike solutions of the quintic complex Ginzburg-Landau equation,” Journal of the Optical Society of America B, vol. 13, no. 7, pp. 1439–1449, 1996.
10. J. M. Soto-Crespo, N. N. Akhmediev, V. V. Afanasjev, and S. Wabnitz, “Pulse solutions of the cubic-quintic complex Ginzburg-Landau equation in the case of normal dispersion,” Physical Review E, vol. 55, no. 4, pp. 4783–4796, 1997.
11. N. Akhmediev, J. M. Soto-Crespo, and G. Town, “Pulsating solitons, chaotic solitons, period doubling, and pulse coexistence in mode-locked lasers: complex Ginzburg-Landau equation approach,” Physical Review E, vol. 63, no. 5, Article ID 056602, 4 pages, 2001.
12. J. M. Soto-Crespo and N. Akhmediev, “Soliton as strange attractor: nonlinear synchronization and chaos,” Physical Review Letters, vol. 95, no. 2, Article ID 024101, 4 pages, 2005. View at Publisher · View at Google Scholar
13. A. Ankiewicz, N. Devine, N. N. Akhmediev, and J. M. Soto-Crespo, “Dissipative solitons and anti-solitons,” Physical Review A, vol. 370, pp. 454–458, 2007.
14. N. Akhmediev, J. M. Soto-Crespo, and P. Grelu, “Roadmap to ultra-short record high-energy pulses out of laser oscillators,” Physics Letters A, vol. 372, no. 17, pp. 3124–3128, 2008. View at Publisher · View at Google Scholar
15. E. Podivilov and V. L. Kalashnikov, “Heavily-chirped solitary pulses in the normal dispersion region: new solutions of the cubic-quintic complex Ginzburg-Landau equation,” JETP Letters, vol. 82, no. 8, pp. 467–471, 2005. View at Publisher · View at Google Scholar
16. V. L. Kalashnikov, E. Podivilov, A. Chernykh, and A. Apolonski, “Chirped-pulse oscillators: theory and experiment,” Applied Physics B, vol. 83, no. 4, pp. 503–510, 2006. View at Publisher · View at Google Scholar
17. V. E. Zakharov and A. B. Shabat, “Exact theory of two-dimensional self-focusing and onedimensional self-modulation of waves in nonlinear media,” Zhurnal Eksperimentalnoi i Teoreticheskoi Fiziki, vol. 61, pp. 118–134, 1971.
18. V. E. Zakharov and A. B. Shabat, “Exact theory of two-dimensional self-focusing and onedimensional self-modulation of waves in nonlinear media,” Soviet Physics Journal of Experimental and Theoretical Physics, vol. 34, pp. 62–69, 1972.
19. A. Hasegawa and F. Tappert, “Transmission of stationary nonlinear optical pulses in dispersive dielectric fibers. I. Anomalous dispersion,” Applied Physics Letters, vol. 23, no. 3, pp. 142–144, 1973. View at Publisher · View at Google Scholar
20. A. Hasegawa and Y. Kodama, Solitons in Optical Communications, Oxford University Press, Oxford, UK, 1995.
21. L. F. Mollenauer and J. P. Gordon, Solitons in Optical Fibers: Fundamentals and Applications, Academic Press, 2006.
22. L. F. Mollenauer, P. V. Mamyshev, J. Gripp, et al., “Experimental demonstration of massive WDM overtransoceanic distances using dispersion managed solitons,” in Massive WDM and TDM Soliton Transmission Systems, A. Hasegawa, Ed., vol. 6, pp. 115–127, Springer, Dordrecht, The Netherlands, 2000.
23. I. Nasieva, J. D. Ania-Castañón, and S. K. Turitsyn, “Nonlinearity management in fibre links with distributed amplification,” Electronics Letters, vol. 39, no. 11, pp. 856–857, 2003. View at Publisher · View at Google Scholar
24. I. R. Gabitov and S. K. Turitsyn, “Averaged pulse dynamics in a cascaded transmission system with passive dispersion compensation,” Optics Letters, vol. 21, no. 5, pp. 327–329, 1996.
25. M. J. Ablowitz and G. Biondini, “Multiscale pulse dynamics in communication systems with strong dispersion management,” Optics Letters, vol. 23, no. 21, pp. 1668–1670, 1998.
26. Y. Chen and H. A. Haus, “Dispersion-managed solitons in the net positive dispersion regime,” Journal of the Optical Society of America B, vol. 16, no. 1, pp. 24–30, 1999.
27. S. K. Turitsyn, E. G. Shapiro, S. B. Medvedev, M. P. Fedoruk, and V. K. Mezentsev, “Physics and mathematics of dispersion-managed optical solitons,” Comptes Rendus Physique, vol. 4, no. 1, pp. 145–161, 2003. View at Publisher · View at Google Scholar
28. S. K. Turitsyn, T. Schäfer, K. H. Spatschek, and V. K. Mezentsev, “Path-averaged chirped optical soliton in dispersion-managed fiber communication lines,” Optics Communications, vol. 163, no. 1, pp. 122–158, 1999. View at Publisher · View at Google Scholar
29. T. I. Lakoba, J. Yang, D. J. Kaup, and B. A. Malomed, “Conditions for stationary pulse propagation in the strong dispersion management regime,” Optics Communications, vol. 149, no. 4–6, pp. 366–375, 1998.
30. I. Gabitov, E. G. Shapiro, and S. K. Turitsyn, “Asymptotic breathing pulse in optical transmission systems with dispersion compensation,” Physical Review E, vol. 55, no. 3, pp. 3624–3633, 1997.
31. T.-S. Yang and W. L. Kath, “Analysis of enhanced-power solitons in dispersion-managed optical fibers,” Optics Letters, vol. 22, no. 13, pp. 985–987, 1997.
32. I. Gabitov, E. G. Shapiro, and S. K. Turitsyn, “Optical pulse dynamics in fiber links with dispersion compensation,” Optics Communications, vol. 134, no. 1–6, pp. 317–329, 1997.
33. N. J. Smith, F. M. Knox, N. J. Doran, K. J. Blow, and I. Bennion, “Enhanced power solitons in optical fibres with periodic dispersion management,” Electronics Letters, vol. 32, no. 1, pp. 54–55, 1996.
34. N. J. Smith, N. J. Doran, F. M. Knox, and W. Forysiak, “Energy-scaling characteristics of solitons in strongly dispersion-managed fibers,” Optics Letters, vol. 21, no. 24, pp. 1981–1983, 1996.
35. A. Berntson, N. J. Doran, W. Forysiak, and J. H. B. Nijhof, “Power dependence of dispersion-managed solitons for anomalous, zero, and normal path-average dispersion,” Optics Letters, vol. 23, no. 12, pp. 900–902, 1998.
36. Y. Kodama, S. Kumar, and A. Maruta, “Chirped nonlinear pulse propagation in a dispersion-compensated system,” Optics Letters, vol. 22, no. 22, pp. 1689–1691, 1997.
37. E. G. Shapiro and S. K. Turitsyn, “Theory of guiding-center breathing soliton propagation in optical communication systems with strong dispersion management,” Optics Letters, vol. 22, no. 20, pp. 1544–1546, 1997.
38. S. K. Turitsyn, T. Schäfer, and V. K. Mezentsev, “Self-similar core and oscillatory tails of a path-averaged chirped dispersion-managed optical pulse,” Optics Letters, vol. 23, no. 17, pp. 1351–1353, 1998.
39. S. K. Turitsyn, T. Schäfer, and V. K. Mezentsev, “Generalized momentum method to describe high-frequency solitary wave propagation in systems with varying dispersion,” Physical Review E, vol. 58, no. 5, pp. R5264–R5267, 1998.
40. J. N. Kutz, P. Holmes, S. G. Evangelides Jr., and J. P. Gordon, “Hamiltonian dynamics of dispersion-managed breathers,” Journal of the Optical Society of America B, vol. 15, no. 1, pp. 87–96, 1998.
41. J. N. Kutz and S. G. Evangelides Jr., “Dispersion-managed breathers with average normal dispersion,” Optics Letters, vol. 23, no. 9, pp. 685–687, 1998.
42. A. V. Mikhailov, “Variationalism and empiriocriticism (Exact and variational approaches to fibre optics equations),” in Optical Solitons: Theoretical Challenges and Industrial Perspectives, V. E. Zakharov and S. Wabnitz, Eds., pp. 63–72, Springer, Berlin, Germany, 1999.
43. B. A. Malomed, “Variational methods in nonlinear fiber optics and related fields,” Progress in Optics, vol. 43, pp. 69–191, 2002.
44. J. P. Gordon and L. F. Mollenauer, “Scheme for the characterization of dispersion-managed solitons,” Optics Letters, vol. 24, no. 4, pp. 223–225, 1999.
45. I. Gabitov, R. Indik, L. Mollenauer, M. Shkarayev, M. Stepanov, and P. M. Lushnikov, “Twin families of bisolitons in dispersion-managed systems,” Optics Letters, vol. 32, no. 6, pp. 605–607, 2007. View at Publisher · View at Google Scholar
46. J. Moeser, I. Gabitov, and C. K. R. T. Jones, “Pulse stabilization by high-order dispersion management,” Optics Letters, vol. 27, no. 24, pp. 2206–2208, 2002.
47. I. R. Gabitov and P. M. Lushnikov, “Nonlinearity management in a dispersion-managed system,” Optics Letters, vol. 27, no. 2, pp. 113–115, 2002.
48. R.-J. Essiambre, B. Mikkelsen, and G. Raybon, “Intra-channel cross-phase modulation and four-wave mixing in high-speed TDM systems,” Electronics Letters, vol. 35, no. 18, pp. 1576–1578, 1999. View at Publisher · View at Google Scholar
49. P. V. Mamyshev and N. A. Mamysheva, “Pulse-overlapped dispersion-managed data transmission and intrachannel four-wave mixing,” Optics Letters, vol. 24, no. 21, pp. 1454–1456, 1999.
50. S. K. Turitsyn, M. P. Fedoruk, and A. Gornakova, “Reduced-power optical solitons in fiber lines with short-scale dispersion management,” Optics Letters, vol. 24, no. 13, pp. 869–871, 1999.
51. S. K. Turitsyn, N. J. Doran, E. G. Turitsyna, E. G. Shapiro, M. P. Fedoruk, and S. B. Medvedev, “Dispersion-managed transmission systems with short-scale dispersion management,” in Massive WDM and TDM Soliton Transmission Systems, M. Shinomiya and A. Hasegawa, Eds., pp. 235–251, Kluwer Academic Publishers, Dordrecht, The Netherlands, 2000.
52. A. H. Liang, H. Toda, and A. Hasegawa, “High-speed soliton transmission in dense periodic fibers,” Optics Letters, vol. 24, no. 12, pp. 799–801, 1999.
53. L. J. Richardson, W. Forysiak, and N. J. Doran, “Dispersion-managed soliton propagation in short-period dispersion maps,” Optics Letters, vol. 25, no. 14, pp. 1010–1012, 2000.
54. M. Shtaif, “Ultrahigh data-rate transmission using a dense dispersion map with two-fold periodicity,” IEEE Photonics Technology Letters, vol. 20, no. 8, pp. 620–622, 2008. View at Publisher · View at Google Scholar
55. O. Y. Schwartz and S. K. Turitsyn, “Multiple-period dispersion-managed solitons,” Physical Review A, vol. 76, no. 4, Article ID 043819, 7 pages, 2007. View at Publisher · View at Google Scholar
56. N. Akhmediev and A. Ankiewicz, Eds., Dissipative Solitons, vol. 661 of Lecture Notes in Physics, Springer, Dordrecht, The Netherlands, 2005.
57. Y. S. Kivshar and G. P. Agarwal, Optical Solitons: From Fibers to Photonic Crystals, Academic Press, 2003.
58. L. F. Mollenauer and R. H. Stolen, “The soliton laser,” Optics Letters, vol. 9, no. 1, pp. 13–15, 1984.
59. K. Tamura, E. P. Ippen, H. A. Haus, and L. E. Nelson, “77-fs pulse generation from a stretched-pulse mode-locked all-fiber ring laser,” Optics Letters, vol. 18, no. 13, pp. 1080–1082, 1993.
60. F. O. Ilday, F. W. Wise, and T. Sosnowski, “High-energy femtosecond stretched-pulse fiber laser with a nonlinear optical loop mirror,” Optics Letters, vol. 27, no. 17, pp. 1531–1533, 2002.
61. F. O. Ilday, J. R. Buckley, W. G. Clark, and F. W. Wise, “Self-similar evolution of parabolic pulses in a laser,” Physical Review Letters, vol. 92, no. 21, Article ID 213902, 4 pages, 2004. View at Publisher · View at Google Scholar
62. F. O. Ilday, J. R. Buckley, H. Lim, F. W. Wise, and W. G. Clark, “Generation of 50-fs, 5-nJ pulses at 1.03 $\mu$m from a wave-breaking-free fiber laser,” Optics Letters, vol. 28, no. 15, pp. 1365–1367, 2003.
63. O. E. Martinez, R. L. Fork, and J. P. Gordon, “Theory of passively mode-locked laser including self-phase modulation and group-velocity dispersion,” Optics Letters, vol. 9, no. 5, pp. 156–158, 1984.
64. S. Namiki, E. P. Ippen, H. A. Haus, and C. X. Yu, “Energy rate equations for mode-locked lasers,” Journal of the Optical Society of America B, vol. 14, no. 8, pp. 2099–2111, 1997.
65. C. Antonelli, J. Chen, and F. X. Kärtner, “Intracavity pulse dynamics and stability for passively mode-locked lasers,” Optics Express, vol. 15, no. 10, pp. 5919–5924, 2007. View at Publisher · View at Google Scholar
66. N. G. Usechak and G. P. Agrawal, “Semi-analytic technique for analyzing mode-locked lasers,” Optics Express, vol. 13, no. 6, pp. 2075–2081, 2005. View at Publisher · View at Google Scholar
67. B. G. Bale and J. N. Kutz, “Variational method for mode-locked lasers,” Journal of the Optical Society of America B, vol. 25, no. 7, pp. 1193–1202, 2008. View at Publisher · View at Google Scholar
68. Y. Chen, F. X. Kärtner, U. Morgner, et al., “Dispersion-managed mode locking,” Journal of the Optical Society of America B, vol. 16, no. 11, pp. 1999–2004, 1999.
69. A. Komarov, H. Leblond, and F. Sanchez, “Quintic complex Ginzburg-Landau model for ring fiber lasers,” Physical Review E, vol. 72, no. 2, Article ID 025604, 4 pages, 2005. View at Publisher · View at Google Scholar
70. H. Leblond, M. Salhi, A. Hideur, T. Chartier, M. Brunel, and F. Sanchez, “Experimental and theoretical study of the passively mode-locked ytterbium-doped double-clad fiber laser,” Physical Review A, vol. 65, Article ID 063811, 8 pages, 2002.
71. F. O. Ilday and F. W. Wise, “Nonlinearity management: a route to high-energy soliton fiber lasers,” Journal of the Optical Society of America B, vol. 19, no. 3, pp. 470–476, 2002.
72. D. Anderson, M. Lisak, and A. Berntson, “A variational approach to nonlinear evolution equations in optics,” Pramana Journal of Physics, vol. 57, no. 5-6, pp. 917–936, 2001.
73. E. N. Tsoy, A. Ankiewicz, and N. Akhmediev, “Dynamical models for dissipative localized waves of the complex Ginzburg-Landau equation,” Physical Review E, vol. 73, no. 3, Article ID 036621, 10 pages, 2006. View at Publisher · View at Google Scholar