Abstract
A brief overview of recent theoretical results in the area of three-dimensional dissipative optical solitons is given. A systematic analysis demonstrates the existence and stability of both fundamental (spinless) and spinning three-dimensional dissipative solitons in both normal and anomalous group-velocity regimes. Direct numerical simulations of the evolution of stationary solitons of the three-dimensional cubic-quintic Ginzburg-Landau equation show full agreement with the predictions based on computation of the instability eigenvalues from the linearized equations for small perturbations. It is shown that the diffusivity in the transverse plane is necessary for the stability of vortex solitons against azimuthal perturbations, while fundamental (zero-vorticity) solitons may be stable in the absence of diffusivity. It has also been found that, at values of the nonlinear gain above the upper border of the soliton existence domain, the three-dimensional dissipative solitons either develop intrinsic pulsations or start to expand in the temporal (longitudinal) direction keeping their structure in the transverse spatial plane.
[1] N. Akhmediev, A. Ankiewicz (Eds.), Dissipative Solitons, Lect. Notes Phys. 661 (Springer, Berlin, 2005) 10.1007/b11728Search in Google Scholar
[2] N.N. Akhmediev, A. Ankiewicz, Solitons: Nonlinear Pulses and Beams (Chapman & Hall, London, 1997) Search in Google Scholar
[3] G. Nicolis, I. Prigogine, Self-organization in nonequilibrium systems (John Wiley and Sons, New York, 1977) Search in Google Scholar
[4] G.I. Stegeman, D.N. Christodoulides, M. Segev, IEEE J. Select. Top. Quant. Electron. 6, 1419 (2000) http://dx.doi.org/10.1109/2944.90219710.1109/2944.902197Search in Google Scholar
[5] Y.S. Kivshar, G.P. Agrawal, Optical Solitons: From Fibers to Photonics Crystals (Academic Press, Amsterdam, 2003) 10.1016/B978-012410590-4/50012-7Search in Google Scholar
[6] B.A. Malomed, D. Mihalache, F. Wise, L. Torner, J. Opt. B: Quantum Semiclassical Opt. 7, R53 (2005) http://dx.doi.org/10.1088/1464-4266/7/5/R0210.1088/1464-4266/7/5/R02Search in Google Scholar
[7] I.S. Aranson, L. Kramer, Rev. Mod. Phys. 74, 99 (2002) http://dx.doi.org/10.1103/RevModPhys.74.9910.1103/RevModPhys.74.99Search in Google Scholar
[8] W.J. Firth, D.V. Skryabin, Phys. Rev. Lett. 79, 2450 (1997) http://dx.doi.org/10.1103/PhysRevLett.79.245010.1103/PhysRevLett.79.2450Search in Google Scholar
[9] L.-C. Crasovan, B.A. Malomed, D. Mihalache, Phys. Rev. E 63, 016605 (2001) http://dx.doi.org/10.1103/PhysRevE.63.01660510.1103/PhysRevE.63.016605Search in Google Scholar
[10] L.-C. Crasovan, B.A. Malomed, D. Mihalache, Phys. Lett. A 289, 59 (2001) http://dx.doi.org/10.1016/S0375-9601(01)00571-010.1016/S0375-9601(01)00571-0Search in Google Scholar
[11] D. Mihalache et al., Phys. Rev. Lett. 97, 073904 (2006) http://dx.doi.org/10.1103/PhysRevLett.97.07390410.1103/PhysRevLett.97.073904Search in Google Scholar PubMed
[12] D. Mihalache, D. Mazilu, F. Lederer, H. Leblond, B.A. Malomed, Phys. Rev. A 75, 033811 (2007) http://dx.doi.org/10.1103/PhysRevA.75.03381110.1103/PhysRevA.75.033811Search in Google Scholar
[13] D. Mihalache, D. Mazilu, L.-C. Crasovan, B.A. Malomed, F. Lederer, Phys. Rev. E 62, R1505 (2000) http://dx.doi.org/10.1103/PhysRevE.62.R150510.1103/PhysRevE.62.R1505Search in Google Scholar
[14] D. Mihalache, D. Mazilu, L.-C. Crasovan, B.A. Malomed, F. Lederer, Phys. Rev. E 61, 7142 (2000) http://dx.doi.org/10.1103/PhysRevE.61.714210.1103/PhysRevE.61.7142Search in Google Scholar
[15] A.V. Buryak, Yu.S. Kivshar, S. Trillo, Opt. Lett. 20, 1961 (1995) 10.1364/OL.20.001961Search in Google Scholar
[16] M.A. Karpierz, Opt. Lett. 20, 1677 (1995) http://dx.doi.org/10.1364/OL.20.00167710.1364/OL.20.001677Search in Google Scholar
[17] I. Towers, A.V. Buryak, R.A. Sammut, B.A. Malomed, Phys. Rev. E 63, 055601(R) (2001) http://dx.doi.org/10.1103/PhysRevE.63.05560110.1103/PhysRevE.63.055601Search in Google Scholar
[18] M. Quiroga-Teixeiro, H. Michinel, J. Opt. Soc. Am. B 14, 2004 (1997) http://dx.doi.org/10.1364/JOSAB.14.00200410.1364/JOSAB.14.002004Search in Google Scholar
[19] I. Towers et al., Phys. Lett. A 288, 292 (2001) http://dx.doi.org/10.1016/S0375-9601(01)00565-510.1016/S0375-9601(01)00565-5Search in Google Scholar
[20] B.A. Malomed, L.-C. Crasovan, D. Mihalache, Physica D 161, 187 (2002) http://dx.doi.org/10.1016/S0167-2789(01)00362-110.1016/S0167-2789(01)00362-1Search in Google Scholar
[21] R.L. Pego, H.A. Warchall, J. Nonlinear Sci. 12, 347 (2002) http://dx.doi.org/10.1007/s00332-002-0475-310.1007/s00332-002-0475-3Search in Google Scholar
[22] T.A. Davydova, A.I. Yakimenko, Yu.A. Zaliznyak, Phys. Rev. E 67, 026402 (2003) http://dx.doi.org/10.1103/PhysRevE.67.02640210.1103/PhysRevE.67.026402Search in Google Scholar PubMed
[23] D. Mihalache, D. Mazilu, B.A. Malomed, F. Lederer, Phys. Rev. E 69, 066614 (2004) http://dx.doi.org/10.1103/PhysRevE.69.06661410.1103/PhysRevE.69.066614Search in Google Scholar PubMed
[24] D. Mihalache, D. Mazilu, B.A. Malomed, F. Lederer, J. Opt. B: Quantum Semiclassical Opt. 6, S341 (2004) http://dx.doi.org/10.1088/1464-4266/6/5/02410.1088/1464-4266/6/5/024Search in Google Scholar
[25] D. Mihalache et al., Phys. Rev. Lett. 88, 073902 (2002) http://dx.doi.org/10.1103/PhysRevLett.88.07390210.1103/PhysRevLett.88.073902Search in Google Scholar PubMed
[26] D. Mihalache et al., Phys. Rev. E 66, 016613 (2002) http://dx.doi.org/10.1103/PhysRevE.66.01661310.1103/PhysRevE.66.016613Search in Google Scholar PubMed
[27] D. Mihalache, D. Mazilu, I. Towers, B.A. Malomed, F. Lederer, Phys. Rev. E 67, 056608 (2003) http://dx.doi.org/10.1103/PhysRevE.67.05660810.1103/PhysRevE.67.056608Search in Google Scholar PubMed
[28] D.V. Skryabin, A.G. Vladimirov, Phys. Rev. Lett. 89, 044101 (2002) http://dx.doi.org/10.1103/PhysRevLett.89.04410110.1103/PhysRevLett.89.044101Search in Google Scholar PubMed
[29] A.S. Desyatnikov, Y.S. Kivshar, Phys. Rev. Lett. 88, 053901 (2002) http://dx.doi.org/10.1103/PhysRevLett.88.05390110.1103/PhysRevLett.88.053901Search in Google Scholar PubMed
[30] Y.V. Kartashov, L.-C. Crasovan, D. Mihalache, L. Torner, Phys. Rev. Lett. 89, 273902 (2002) http://dx.doi.org/10.1103/PhysRevLett.89.27390210.1103/PhysRevLett.89.273902Search in Google Scholar PubMed
[31] L.-C. Crasovan et al., Phys. Rev. E 67, 046610 (2003) http://dx.doi.org/10.1103/PhysRevE.67.04661010.1103/PhysRevE.67.046610Search in Google Scholar PubMed
[32] D. Mihalache et al., Phys. Rev. E 68, 046612 (2003) http://dx.doi.org/10.1103/PhysRevE.68.04661210.1103/PhysRevE.68.046612Search in Google Scholar PubMed
[33] D. Mihalache et al., J. Opt. B: Quantum Semiclassical Opt. 6, S333 (2004) http://dx.doi.org/10.1088/1464-4266/6/5/02310.1088/1464-4266/6/5/023Search in Google Scholar
[34] P. Grelu, J.M. Soto-Crespo, N. Akhmediev, Opt. Express 13, 9352 (2005) http://dx.doi.org/10.1364/OPEX.13.00935210.1364/OPEX.13.009352Search in Google Scholar
[35] J.M. Soto-Crespo, P. Grelu, N. Akhmediev, Opt. Express 14, 4013 (2006) http://dx.doi.org/10.1364/OE.14.00401310.1364/OE.14.004013Search in Google Scholar
[36] V. Skarka, N.B. Aleksic, Phys. Rev. Lett. 96, 013903 (2006) http://dx.doi.org/10.1103/PhysRevLett.96.01390310.1103/PhysRevLett.96.013903Search in Google Scholar
[37] H. Sakaguchi, B.A. Malomed, Physica D 167, 123 (2002) http://dx.doi.org/10.1016/S0167-2789(02)00532-810.1016/S0167-2789(02)00532-8Search in Google Scholar
[38] S. Skupin et al., Phys. Rev. E 70, 046602 (2004) http://dx.doi.org/10.1103/PhysRevE.70.04660210.1103/PhysRevE.70.046602Search in Google Scholar PubMed
© 2008 Versita Warsaw
This work is licensed under the Creative Commons Attribution-NonCommercial-NoDerivatives 3.0 License.