[1] Zakharov, V. E., L'vov, V. S. & Falkovich, G. Kolmogorov Spectra of Turbulence I: Wave Turbulence. (Springer-Verlag, Berlin, 1992).
[2] Staliunas, K. & Sánchez-Morcillo, V. J. Transverse Patterns in Nonlinear Optical Resonators. (Springer: Berlin, 2003).
[3] Lugiato, L., Prati, F. & Brambilla, M. Nonlinear Optical Systems. (Cambridge University Press, Cambridge, 2015).
[4] Cross, M. C. & Hohenberg, P. C. Pattern formation outside of equilibrium. Rev. Mod. Phys. 65, 851–1112 (1993). doi: 10.1103/RevModPhys.65.851
[5] Aranson, I. S. & Kramer, L. The world of the complex Ginzburg-Landau equation. Rev. Mod. Phys. 74, 99–143 (2002). doi: 10.1103/RevModPhys.74.99
[6] Agrawal, G. P. Nonlinear Fiber Optics. 4th edn, (Academic Press, San Diego, 2006).
[7] Ablowitz, M. J. Nonlinear Dispersive Waves: Asymptotic Analysis and Solitons.. (Cambridge University Press, Cambridge, 2011).
[8] Zakharov, V. E. & Ostrovsky, L. A. Modulation instability: the beginning. Physica D 238, 540–548 (2009). doi: 10.1016/j.physd.2008.12.002
[9] Benjamin, T. B. & Feir, J. E. The disintegration of wave trains on deep water part 1. Theory. J. Fluid. Mech. 27, 417–430 (1967).
[10] Bespalov, V. I. & Talanov, V. I. Filamentary structure of light beams in nonlinear liquids. JETP Lett. 3, 307–311 (1966).
[11] Tai, K., Hasegawa, A. & Tomita, A. Observation of modulational instability in optical fibers. Phys. Rev. Lett. 56, 135–138 (1986). doi: 10.1103/PhysRevLett.56.135
[12] Agrawal, G. P. Modulation instability induced by cross-phase modulation. Phys. Rev. Lett. 59, 880–883 (1987). doi: 10.1103/PhysRevLett.59.880
[13] Berkhoer, A. L. & Zakharov, V. E. Self-excitation of waves with different polarizations in nonlinear media. Sov. Phys. JETP 31, 486–490 (1970).
[14] Wabnitz, S. Modulational polarization instability of light in a nonlinear birefringent dispersive medium. Phys. Rev. A 38, 2018–2021 (1988). doi: 10.1103/PhysRevA.38.2018
[15] Trillo, S., Wabnitz, S., Stolen, R. H., Assanto, G. & Seaton, C. T. et al. Experimental observation of polarization instability in a birefringent optical fiber. Appl. Phys. Lett. 49, 1224–1226 (1986). doi: 10.1063/1.97420
[16] Agrawal, G. P. Modulation instability in Erbium-doped fiber amplifiers. IEEE Photon. Technol. Lett. 4, 562–564 (1992). doi: 10.1109/68.141968
[17] Van Tartwijk, G. H. M. & Agrawal, G. P. Maxwell-Bloch dynamics and modulation instabilities in fiber lasers and amplifiers. J. Opt. Soc. Am. B 14, 2618–2627 (1997). doi: 10.1364/JOSAB.14.002618
[18] Turitsyn, S., Rubenchik, A. M. & Fedoruk, M. P. On the theory of the modulation instability in optical fiber amplifiers. Opt. Lett. 35, 2864–2866 (2010). doi: 10.1364/OL.35.002684
[19] Dudley, J. M., Goënty, G. & Coen, S. Supercontinuum generation in photonic crystal fiber. Rev. Mod. Phys. 78, 1135–1184 (2006). doi: 10.1103/RevModPhys.78.1135
[20] Kip, D., Soljacic, M., Segev, M., Eugenieva, E. & Christodoulides, D. N. Modulation instability and pattern formation in spatially incoherent light beams. Science 290, 495–498 (2000). doi: 10.1126/science.290.5491.495
[21] Zakharov, V. E. & Shabat, A. B. Exact theory of two-dimensional self-focussing and one-dimensional self-modulation of waves in nonlinear media. Sov. Phys. JETP 34, 62–69 (1972).
[22] Kuznetsov, E. Solitons in a parametrically unstable plasma. Sov. Phys. Dokl. 22, 507–508 (1977).
[23] Akhmediev, N. & Korneev, V. I. Modulation instability and periodic solutions of the nonlinear Schrödinger equation. Theor. Math. Phys. 69, 1089–1093 (1986). doi: 10.1007/BF01037866
[24] Peregrine, D. H. Water waves, nonlinear Schrödinger equations and their solutions. ANZIAM J. 25, 16–43 (1983).
[25] Dudley, J. M., Dias, F., Erkintalo, M. & Genty, G. Instabilities, breathers and rogue waves in optics. Nat. Photon. 8, 755–764 (2014). doi: 10.1038/nphoton.2014.220
[26] Faraday, M. On a peculiar class of acoustical figures; and on certain forms assumed by groups of particles upon vibrating elastic surfaces. Philos. Trans. R. Soc. Lond. 121, 299–340 (1831). doi: 10.1098/rstl.1831.0018
[27] Benjamin, T. B. & Ursell, F. The stability of the plane free surface of a liquid in vertical periodic motion. Proc. R. Soc. A 225, 505–515 (1954). doi: 10.1098/rspa.1954.0218
[28] Coullet, P., Frisch, T. & Sonnino, G. Dispersion-induced patterns. Phys. Rev. E 49, 2087–2090 (1994). doi: 10.1103/PhysRevE.49.2087
[29] Petrov, V., Ouyang, Q. & Swinney, H. L. Resonant pattern formation in a chemical system. Nature 388, 655–657 (1997). doi: 10.1038/41732
[30] Melo, F., Umbanhowar, P. & Swinney, H. L. Transition to parametric wave patterns in a vertically oscillated granular layer. Phys. Rev. Lett. 72, 172–175 (1994). doi: 10.1103/PhysRevLett.72.172
[31] Drake, J. F., Kaw, P. K., Lee, Y. C., Schmid, G. & Liu, C. S. et al. Parametric instabilities of electromagnetic waves in plasmas. Phys. Fluids 17, 778–785 (1974). doi: 10.1063/1.1694789
[32] Staliunas, K., Longhi, S. & De Valcárcel, G. J. Faraday patterns in Bose-Einstein condensates. Phys. Rev. Lett. 89, 210406 (2002). doi: 10.1103/PhysRevLett.89.210406
[33] Engels, P., Atherton, C. & Hoefer, M. A. Observation of Faraday waves in a Bose-Einstein condensate. Phys. Rev. Lett. 98, 095301 (2007). doi: 10.1103/PhysRevLett.98.095301
[34] Centurion, M., Porter, M. A., Pu, Y., Kevrekidis, P. G. & Frantzeskakis, D. J. et al. Modulational instability in a layered Kerr medium: theory and experiment. Phys. Rev. Lett. 97, 234101 (2006). doi: 10.1103/PhysRevLett.97.234101
[35] Szwaj, C., Bielawski, S., Derozier, D. & Erneux, T. Faraday instability in a multimode laser. Phys. Rev. Lett. 80, 3968 (1998). doi: 10.1103/PhysRevLett.80.3968
[36] Matera, F., Mecozzi, A., Romagnoli, M. & Settembre, M. Sideband instability induced by periodic power variation in long-distance fiber links. Opt. Lett. 18, 1499–1501 (1993). doi: 10.1364/OL.18.001499
[37] Abdullaev, F. K., Darmanyan, S. A., Bischoff, S. & Sørensen, M. P. Modulational instability of electromagnetic waves in media with varying nonlinearity. J. Opt. Soc. Am. B 14, 27–33 (1997). doi: 10.1364/JOSAB.14.000027
[38] Smith, N. J. & Doran, N. J. Modulational instabilities in fibers with periodic dispersion management. Opt. Lett. 21, 570–572 (1996). doi: 10.1364/OL.21.000570
[39] Abdullaev, F. K., Darmanyan, S. A., Kobyakov, A. & Lederer, F. Modulational instability in optical fibers with variable dispersion. Phys. Lett. A 220, 213–218 (1996). doi: 10.1016/0375-9601(96)00504-X
[40] Bronski, J. C. & Kutz, J. N. Modulational stability of plane waves in nonreturn-to-zero communications systems with dispersion management. Opt. Lett. 21, 937–939 (1996). doi: 10.1364/OL.21.000937
[41] Mussot, A., Conforti, M., Trillo, S., Copie, F. & Kudlinski, A. Modulation instability in dispersion oscillating fibers. Adv. Opt. Photon. 10, 1–42 (2018). doi: 10.1364/AOP.10.000001
[42] Staliunas, K., Hang, C. & Konotop, V. V. Parametric patterns in optical fiber ring nonlinear resonators. Phys. Rev. A 88, 023846 (2013). doi: 10.1103/PhysRevA.88.023846
[43] Conforti, M., Mussot, A., Kudlinski, A. & Trillo, S. Modulational instability in dispersion oscillating fiber ring cavities. Opt. Lett. 39, 4200–4203 (2014). doi: 10.1364/OL.39.004200
[44] Turing, A. M. The chemical basis of morphogenesis. Philos. Trans. R. Soc. B 237, 37–72 (1952). doi: 10.1098/rstb.1952.0012
[45] Lugiato, L. A. & Lefever, R. Spatial dissipative structures in passive optical systems. Phys. Rev. Lett. 58, 2209–2211 (1987). doi: 10.1103/PhysRevLett.58.2209
[46] Copie, F., Conforti, M., Kudlinski, A., Mussot, A., & Trillo, S. Competing Turing and Faraday instabilities in longitudinally modulated passive resonators. Phys. Rev. Lett. 116, 143901 (2016). doi: 10.1103/PhysRevLett.116.143901
[47] Staliunas, K. & Sánchez-Morcillo, V. J. Turing patterns in nonlinear optics. Opt. Commun. 177, 389–395 (2000). doi: 10.1016/S0030-4018(00)00561-7
[48] Haelterman, M., Trillo, S. & Wabnitz, S. Additive-modulation-instability ring laser in the normal dispersion regime of a fiber. Opt. Lett. 17, 745–747 (1992). doi: 10.1364/OL.17.000745
[49] Coen, S. & Haelterman, M. Modulational instability induced by cavity boundary conditions in a normally dispersive optical fiber. Phys. Rev. Lett. 79, 4139–4142 (1997). doi: 10.1103/PhysRevLett.79.4139
[50] Perego, A. M., Smirnov, S. V., Staliunas, K., Churkin, D. V. & Wabnitz, S. Self-induced Faraday instability laser. Phys. Rev. Lett. 120, 213902 (2018). doi: 10.1103/PhysRevLett.120.213902
[51] Guasoni, M. Generalized modulational instability in multimode fibers: wideband multimode parametric amplification. Phys. Rev. A 92, 033849 (2015). doi: 10.1103/PhysRevA.92.033849
[52] Longhi, S. Modulational instability and space–time dynamics in nonlinear parabolic-index optical fibers. Opt. Lett. 28, 2363–2365 (2003). doi: 10.1364/OL.28.002363
[53] Krupa, K., Tonello, A., Barthélémy, A., Couderc, V. & Shalaby, B. M. et al. Observation of geometric parametric instability induced by the periodic spatial self-imaging of multimode waves. Phys. Rev. Lett. 116, 183901 (2016). doi: 10.1103/PhysRevLett.116.183901
[54] Wright, L. G., Liu, Z. W., Nolan, D. A., Li, M. J. & Christodoulides, D. N. et al. Self-organized instability in graded-index multimode fibres. Nat. Photon. 10, 771–776 (2016). doi: 10.1038/nphoton.2016.227
[55] Haken, H. Analogy between higher instabilities in fluids and lasers. Phys. Lett. A 53, 77–78 (1975). doi: 10.1016/0375-9601(75)90353-9
[56] Arecchi, F. T., Giacomelli, G., Ramazza, P. L. & Residori, S. Experimental evidence of chaotic itinerancy and spatiotemporal chaos in optics. Phys. Rev. Lett. 65, 2531–2534 (1990). doi: 10.1103/PhysRevLett.65.2531
[57] Risken, H. & Nummedal, K. Instability of off resonance modes in lasers. Phys. Lett. A 26, 275–276 (1968). doi: 10.1016/0375-9601(68)90646-4
[58] Graham, R. & Haken, H. Quantum theory of light propagation in a fluctuating laser-active medium. Z. Phys. A 213, 420–450 (1968). doi: 10.1007/BF01405384
[59] Pessina, E. M., Bonfrate, G., Fontana, F. & Lugiato, L. A. Experimental observation of the Risken-Nummedal-Graham-Haken multimode laser instability. Phys. Rev. A 56, 4086–4093 (1997). doi: 10.1103/PhysRevA.56.4086
[60] Nakazawa, M., Suzuki, K. & Haus, H. A. The modulational instability laser. I. Experiment. IEEE J. Quant. Electron. 25, 2036–2044 (1989).
[61] Boyd, R. W. Nonlinear Optics. (Academic Press, San Diego, 1991).
[62] Armstrong, J. A., Bloembergen, N., Ducuing, J. & Pershan, P. S. Interactions between light waves in a nonlinear dielectric. Phys. Rev. 127, 1918–1939 (1962). doi: 10.1103/PhysRev.127.1918
[63] Cappellini, G. & Trillo, S. Third-order three-wave mixing in single-mode fibers: exact solutions and spatial instability effects. J. Opt. Soc. Am. B 8, 824–838 (1991). doi: 10.1364/JOSAB.8.000824
[64] Ott, J. R., Steffensen, H., Rottwitt, K. & McKinstrie, C. J. Geometric interpretation of four-wave mixing. Phys. Rev. A 88, 043805 (2013). doi: 10.1103/PhysRevA.88.043805
[65] Stolen, R. H. Phase-matched-stimulated four-photon mixing in silica-fiber waveguides. IEEE J. Quant. Electron. 11, 100–103 (1975). doi: 10.1109/JQE.1975.1068571
[66] Marhic, M. Fiber Optical Parametric Amplifiers, Oscillators and Related Devices. (Cambridge University Press, Cambridge, 2007).
[67] Torounidis, T., Andrekson, P. A. & Olsson, B. E. Fiber-optical parametric amplifier with 70-dB gain. IEEE Photon. Technol. Lett. 18, 1194–1196 (2006). doi: 10.1109/LPT.2006.874714
[68] Stephens, M. F. C., Phillips, I. D., Rosa, P., Harper, P. & Doran, N. J. Improved WDM performance of a fibre optical parametric amplifier using Raman-assisted pumping. Opt. Exp. 23, 902–911 (2015). doi: 10.1364/OE.23.000902
[69] Hansryd, J., Andrekson, P. A., Westlund, M., Li, J. & Hedekvist, P. O. Fiber-based optical parametric amplifiers and their applications. IEEE J. Sel. Top. Quant. Electron. 8, 506–520 (2002). doi: 10.1109/JSTQE.2002.1016354
[70] Radic, S. & McKinstrie, C. J. Two-pump fiber parametric amplifiers. Opt. Fiber Technol. 9, 7–23 (2003). doi: 10.1016/S1068-5200(02)00528-X
[71] McKinstrie, C. J. & Radic, S. Phase-sensitive amplification in a fiber. Opt. Exp. 12, 4973–4979 (2004). doi: 10.1364/OPEX.12.004973
[72] Marhic, M. E., Andrekson, P. A., Petropoulos, P., Radic, S. & Peucheret, C. et al. Fiber optical parametric amplifiers in optical communication systems. Laser Photon. Rev. 9, 50–74 (2015). doi: 10.1002/lpor.201400087
[73] Barthelemy, A. & De La Fuente, R. Unusual modulation instability in fibers with normal and anomalous dispersions. Opt. Commun. 73, 409–412 (1989). doi: 10.1016/0030-4018(89)90181-8
[74] Turitsyn, S. K., Bednyakova, A. E., Fedoruk, M. P., Papernyi, S. B. & Clements, W. R. L. Inverse four-wave mixing and self-parametric amplification in optical fibre. Nat. Photon. 9, 608–614 (2015). doi: 10.1038/nphoton.2015.150
[75] Picozzi, A., Garnier, J., Hansson, T., Suret, P. & Randoux, S. et al. Optical wave turbulence: towards a unified nonequilibrium thermodynamic formulation of statistical nonlinear optics. Phys. Rep. 542, 1–132 (2014). doi: 10.1016/j.physrep.2014.03.002
[76] Hall, B., Lisak, M., Anderson, D., Fedele, R. & Semenov, V. E. Statistical theory for incoherent light propagation in nonlinear media. Phys. Rev. E 65, 035602 (2002). doi: 10.1103/PhysRevE.65.035602
[77] Churkin, D. V., Kolokolov, I. V., Podivilov, E. V., Vatnik, I. D., & Nikulin, M. A. et al. Wave kinetics of random fibre lasers. Nat. Commun. 2, 6214 (2015). doi: 10.1038/ncomms7214
[78] Babin, S. A., Churkin, D. V., Ismagulov, A. E., Kablukov, S. I. & Podivilov, E. V. Four-wave-mixing-induced turbulent spectral broadening in a long Raman fiber laser. J. Opt. Soc. Am. B 24, 1729–1738 (2007). doi: 10.1364/JOSAB.24.001729
[79] Deng, L., Hagley, E. W., Wen, J., Trippenbach, M. & Band, Y. et al. Four-wave mixing with matter waves. Nature 398, 218–220 (1999). doi: 10.1038/18395
[80] Trippenbach, M., Band, Y. B. & Julienne, P. S. Theory of four-wave mixing of matter waves from a Bose-Einstein condensate. Phys. Rev. A 62, 023608 (2000). doi: 10.1103/PhysRevA.62.023608
[81] Vogels, J. M., Xu, K. & Ketterle, W. Generation of macroscopic pair-correlated atomic beams by four-wave mixing in Bose-Einstein condensates. Phys. Rev. Lett. 89, 020401 (2002). doi: 10.1103/PhysRevLett.89.020401
[82] Karlsson, M. Modulational instability in lossy optical fibers. J. Opt. Soc. Am. B 12, 2071–2077 (1995). doi: 10.1364/JOSAB.12.002071
[83] Krechetnikov, R. & Marsden, J. E. Dissipation-induced instabilities in finite dimensions. Rev. Mod. Phys. 79, 519–553 (2007). doi: 10.1103/RevModPhys.79.519
[84] Tanemura, T., Ozeki, Y. & Kikuchi, K. Modulational instability and parametric amplification induced by loss dispersion in optical fibers. Phys. Rev. Lett. 93, 163902 (2004). doi: 10.1103/PhysRevLett.93.163902
[85] Trillo, S. & Wabnitz, S. Dynamics of the nonlinear modulational instability in optical fibers. Opt. Lett. 16, 986–988 (1991). doi: 10.1364/OL.16.000986
[86] Xu, K., Liu, H. Y., Dai, Y. T., Wu, J. & Lin, J. T. Synthesis of broadband and flat parametric gain by idler loss in optical fiber. Opt. Commun. 285, 790–794 (2012). doi: 10.1016/j.optcom.2011.11.010
[87] Coen, S., Wardle, D. A. & Harvey, J. D. Observation of non-phase-matched parametric amplification in resonant nonlinear optics. Phys. Rev. Lett. 89, 273901 (2002). doi: 10.1103/PhysRevLett.89.273901
[88] Sylvestre, T., Maillotte, H., Lantz, E. & Tchofo Dinda, P. Raman-assisted parametric frequency conversion in a normally dispersive single-mode fiber. Opt. Lett. 15, 1561–1563 (1999). doi: 10.1364/OL.24.001561
[89] Castelli, F., Brambilla, M., Gatti, A., Prati, F. & Lugiato, L. A. The LLE, pattern formation and a novel coherent source. Eur. Phys. J. D. 71, 84 (2017). doi: 10.1140/epjd/e2017-70754-1
[90] Myers, L. E., Eckardt, R. C., Fejer, M. M., Byer, R. L. & Bosenberg, W. R. et al. Quasi-phase-matched optical parametric oscillators in bulk periodically poled LiNbO3. J. Opt. Soc. Am. B 12, 2102–2116 (1995). doi: 10.1364/JOSAB.12.002102
[91] Lowenthal, D. D. CW periodically poled LiNbO3 optical parametric oscillator model with strong idler absorption. IEEE J. Quant. Electron. 34, 1356–1366 (1998). doi: 10.1109/3.704319
[92] Rustad, G., Arisholm, G. & Farsund, Ø. Effect of idler absorption in pulsed optical parametric oscillators. Opt. Exp. 19, 2815–2830 (2011). doi: 10.1364/OE.19.002815
[93] Jauregui, C., Steinmetz, A., Limpert, J. & Tünnermann, A. High-power efficient generation of visible and mid-infrared radiation exploiting four-wave-mixing in optical fibers. Opt. Exp. 20, 24957-24965 (2012).
[94] Ma, J. G., et al. Quasi-parametric amplification of chirped pulses based on a Sm3+-doped yttrium calcium oxyborate crystal. Optica 20, 1006–1009 (2015). doi: 10.1364/OPTICA.2.001006
[95] Perego, A. M., Tarasov, N., Churkin, D. V., Turitsyn, S. K. & Staliunas, K. Pattern generation by dissipative parametric instability. Phys. Rev. Lett. 116, 028701 (2016). doi: 10.1103/PhysRevLett.116.028701
[96] Tarasov, N., Perego, A. M., Churkin, D. V., Staliunas, K. & Turitsyn, S. K. Mode-locking via dissipative Faraday instability. Nat. Commun. 7, 12441 (2016). doi: 10.1038/ncomms12441
[97] Perego, A. M. High-repetition-rate, multi-pulse all-normal-dispersion fiber laser. Opt. Lett. 42, 3574–3577 (2017). doi: 10.1364/OL.42.003574
[98] Staliunas, K. Removal of excitations of Bose-Einstein condensates by space- and time-modulated potentials. Phys. Rev. A. 84, 013626 (2011). doi: 10.1103/PhysRevA.84.013626