# American Institute of Mathematical Sciences

October  2011, 4(5): 923-955. doi: 10.3934/dcdss.2011.4.923

## Nonlinear wave dynamics: From lasers to fluids

 1 Department of Applied Mathematics, University of Colorado, Boulder, CO 80309-0526, United States, United States, United States, United States 2 Department of Mathematics, University of Ioannina, Ioannina 45110, Greece

Received  September 2009 Revised  December 2009 Published  December 2010

Nonlinear wave equations are central to the study of nonlinear optics and fluid dynamics. Notably, recent research has shown that solitons can be generated in mode-locked lasers. An interesting application of these lasers is the development of optical clocks which have the potential to be considerably more accurate than atomic clocks. Another important area of research in nonlinear optics is lattice dynamics where localized solitary wave or solitons can be obtained in periodic and irregular lattice systems. In honeycomb lattices, discrete and continuous nonlinear Dirac systems can be derived in certain parameter regimes; the Dirac systems describe conical diffraction, a phenomena observed in recent experiments. In water and internal waves the classical equations are reformulated as a system of coupled equations where the free surface equations are formulated as nonlocal equation and the depth variable is eliminated. These systems reduce to interesting asymptotic equations in suitable limits. A numerical method, termed spectral renormalization, is used to find solitary waves in nonlinear optics, water waves and multi-fluid systems.
Citation: Mark J. Ablowitz, Terry S. Haut, Theodoros P. Horikis, Sean D. Nixon, Yi Zhu. Nonlinear wave dynamics: From lasers to fluids. Discrete and Continuous Dynamical Systems - S, 2011, 4 (5) : 923-955. doi: 10.3934/dcdss.2011.4.923
##### References:
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##### References:
 [1] M. J. Ablowitz and G. Biondini, Multiscale pulse dynamics in communication systems with strong dispersion management, Opt. Lett., 23 (1998), 1668-1670. doi: 10.1364/OL.23.001668. [2] M. J. Ablowitz, G. Biondini and E. Olson, On the evolution and interaction of dispersion-managed solitons, in "Massive WDM and TDM Soliton Transmission Systems" (Kyoto, Japan), Kluwer Academic Publishers, (2000), 362-367. [3] M. J. Ablowitz, A. S. Fokas and Z. Musslimani, On a new non-local formulation of water waves, J. Fluid Mech., 562 (2006), 313-343. doi: 10.1017/S0022112006001091. [4] M. J. Ablowitz and T. S. Haut, Asymptotic expansions for solitary gravity-capillary waves in two and three dimensions, Proc. Roy. Soc. A, 465 (2009), 2725-2749. doi: 10.1098/rspa.2009.0112. [5] M. J. Ablowitz and T. P. Horikis, Pulse dynamics and solitons in mode-locked lasers, Phys. Rev. A, 78 (2008), 011802(R). doi: 10.1103/PhysRevA.78.011802. [6] M. J. Ablowitz and T. P. Horikis, Solitons and spectral renormalization methods in nonlinear optics, Eur. Phys. J. Special Topics, 173 (2009), 147-166. doi: 10.1140/epjst/e2009-01072-0. [7] M. J. Ablowitz and T. P. Horikis, Solitons in normally dispersive mode-locked lasers, Phys. Rev. A, 79 (2009), 063845. doi: 10.1103/PhysRevA.79.063845. [8] M. J. Ablowitz, T. P. Horikis and B. Ilan, Solitons in dispersion-managed mode-locked lasers, Phys. Rev. A, 77 (2008), 033814. doi: 10.1103/PhysRevA.77.033814. [9] M. J. Ablowitz, T. P. Horikis and S. D. Nixon, Soliton strings and interactions in mode-locked lasers, Opt. Comm., 282 (2009), 4127-4135. doi: 10.1016/j.optcom.2009.07.005. [10] M. J. Ablowitz, T. P. Horikis, S. D. Nixon and Y. Zhu, Asymptotic analysis of pulse dynamics in mode-locked lasers, Stud. Appl. Math., 122 (2009), 411-425. doi: 10.1111/j.1467-9590.2009.00441.x. [11] M. J. Ablowitz, B. Ilan, E. Schonbrun and R. Piestun, Solitons in two-dimensional lattices possessing defects, dislocations, and quasicrystal structures, Phys. Rev. E, 74 (2006), 023623. [12] M. J. Ablowitz and Z. Musslimani, Discrete spatial solitons in a diffraction-managed nonlinear waveguide array: A unified approach, Physica D, 184 (2003), 276-303. doi: 10.1016/S0167-2789(03)00226-4. [13] M. J. Ablowitz and Z. Musslimani, Spectral renormalization method for computing self-localized solutions to nonlinear systems, Opt. Lett., 30 (2005), 2140-2142. doi: 10.1364/OL.30.002140. [14] M. J. Ablowitz, S. D. Nixon and Y. Zhu, Conical diffraction in honeycomb lattices, Phys. Rev. A, 79 (2009), 053830. doi: 10.1103/PhysRevA.79.053830. [15] M. J. Ablowitz and H. Segur, "Solitons and the Inverse Scattering Transform," SIAM Publications, Philadelphia, 1981. [16] G. P. Agrawal, "Nonlinear Fiber Optics," Academic Press, New York, 2001. [17] N. N. Akhmediev and A. Ankiewicz, "Solitons, Nonlinear Pulses and Beams," Chapman & Hall, 1997. [18] N. 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, Phys. Rev. E, 63 (2001), 056602. doi: 10.1103/PhysRevE.63.056602. [19] D. J. Benney and J. C. Luke, On the interactions of permanent waves of finite amplitude, J. Math. Phys., 43 (1964), 309-313. [20] M. V. Berry and M. R. Jeffrey, Conical diffraction: Hamilton's diabolical point at the heart of crystal optics, Progress in Optics, 50 (2007), 13-50. doi: 10.1016/S0079-6638(07)50002-8. [21] J. Boussinesq, Essai sur la théorie des eaux courantes, Mémoires Présentés par Divers Savants à l'Académie des Sciences, 1 (1877), 1-680. [22] R. W. Boyd, "Nonlinear Optics," Elsevier, London, 2002. [23] R. Carretero-González, D. J. Frantzeskakis and P. G. Kevrekidis, Nonlinear waves in Bose-Einstein condensates: Physical relevance and mathematical techniques, Nonlinearity, 21 (2008), R139-R202. doi: 10.1088/0951-7715/21/7/R01. [24] Z. G. Chen, A. Bezryadina, I. Makasyuk and J. K. Yang, Observation of two-dimensional lattice vector solitons, Opt. Lett., 29 (2004), 1656-1658. doi: 10.1364/OL.29.001656. [25] A. Chong, W. H. Renninger and F. W. Wise, Observation of antisymmetric dispersion-managed solitons in a mode-locked laser, Opt. Lett., 33 (2008), 1717-1719. doi: 10.1364/OL.33.001717. [26] A. Chong, W. H. Renninger and F. W. Wise, Properties of normal-dispersion femtosecond fiber lasers, J. Opt. Soc. Am. B, 25 (2008), 140-148. doi: 10.1364/JOSAB.25.000140. [27] D. N. Christodoulides and R. Joseph, Discrete self-focusing in nonlinear arrays of coupled wave-guides, Opt. Lett., 13 (1988), 794-796. doi: 10.1364/OL.13.000794. [28] W. Craig and M. D. Groves, Hamiltonian long-wave approximations to the water-wave problem, Wave Motion, 19 (1994), 367-389. doi: 10.1016/0165-2125(94)90003-5. [29] W. Craig and C. Sulem, Numerical simulation of gravity waves, J. Comp. Phys., 108 (1993), 73-83. doi: 10.1006/jcph.1993.1164. [30] S. Cundiff, J. Ye and J. Hall, Rulers of light, Scientific American, 298 (2008), 74-81. doi: 10.1038/scientificamerican0408-74. [31] S. T. Cundiff, Soliton dynamics in mode-locked lasers, Lect. Notes Phys., 661 (2005), 183-206. doi: 10.1007/10928028_8. [32] S. T. Cundiff, J. M. Soto-Crespo and N. N. Akhmediev, Experimental evidence for soliton explosions, Phys. Rev. Lett., 88 (2002), 073903. doi: 10.1103/PhysRevLett.88.073903. [33] J. W. Dold, An efficient surface-integral algorithm applied to unsteady gravity-waves, J. Comp. Phys., 103 (1992), 90-115. doi: 10.1016/0021-9991(92)90327-U. [34] N. K. Efremidis, J. Hudock, D. N. Christodoulides, J. W. Fleischer, O. Cohen and M. Segev, Two-dimenional optical lattice solitons, Phys. Rev. Lett., 91 (2003), 213906. doi: 10.1103/PhysRevLett.91.213906. [35] N. K. Efremidis, J. Hudock, D. N. Christodoulides, J. W. Fleischer, O. Cohen and M. Segev, Two-dimensional optical lattice solitons, Phys. Rev. Lett., 91 (2003), 213906. doi: 10.1103/PhysRevLett.91.213906. [36] N. K. Efremidis, S. Sears, D. N. Christodoulides, J. W. Fleischer and M. Segev, Discrete solitons in photorefractive optically induced photonic lattices, Phys. Rev. E, 66 (2002), 046602. doi: 10.1103/PhysRevE.66.046602. [37] H. Eisenberg, Y. Silberberg, R. Morandotti, A. Boyd and J. Aitchison, Wannier functions analysis of the nonlinear Schrödinger equation with a periodic potential, Phys. Rev. Lett., 81 (1998), 3382-3386. [38] J. Fenton, A ninth-order solution for the solitary wave, J. Fluid Mech., 53 (1972), 257-271. doi: 10.1017/S002211207200014X. [39] J. W. Fleischer, T. Carmon, M. Segev, N. K. Efremidis and D. N. 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