Addressed here is the occurrence of point singularities which owe to the focusing of short or long waves, a phenomenon labeled dispersive blow-up. The context of this investigation is linear and nonlinear, strongly dispersive equations or systems of equations. The present essay deals with linear and nonlinear Schrödinger equations, a class of fractional order Schrödinger equations and the linearized water wave equations, with and without surface tension. Commentary about how the results may bear upon the formation of rogue waves in fluid and optical environments is also included.
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Abramowitz, M. and Stegun, I. A., Handbook of Mathematical Functions, Dover, New York, 1964.
Akhmediev, N., Soto-Crespo, J. M. and Ankiewicz, A., How to excite a rogue wave, Phy. Rev. A, 80, 2009, 043818, 1–17.
Ben-Artzi, M. and Saut, J.-C., Uniform decay estimates for a class of oscillatory integrals and applications, Diff. Int. Eqs., 12, 1999, 137–145.
Benjamin, T. B. Bona, J. L. and Mahony, J. J., Model equations for long waves in nonlinear, dispersive media, Philos. Trans. Royal Soc. London, Ser. A, 272, 1972, 47–78.
Bona, J. L. and Chen, H. Q., Well-posedness for regularized dispersive wave equations, Disc. Cont. Dyn. Systems A, 23, 2009, 1253–1275.
Bona, J. L., Chen, M. and Saut, J.-C., Boussinesq equations and other systems for small-amplitude long waves in nonlinear dispersive media. I: Derivation and linear theory, J. Nonlinear Sci., 12, 2002, 283–318.
Bona, J. L., Chen, M. and Saut, J.-C., Boussinesq equations and other systems for small-amplitude long waves in nonlinear dispersive media. II: The nonlinear theory, Nonlinearity, 17, 2004, 925–952.
Bona, J. L. and Saut, J.-C., Dispersive blow-up of solutions of generalized Korteweg-de Vries equations, J. Diff. Eqs., 103, 1993, 3–57.
Bona, J. L. and Tzvetkov, N., Sharp well-posedness results for the BBM equation, Disc. Cont. Dyn. Systems A, 23, 2009, 1241–1252.
Brenner, P., The Cauchy problem for systems in l p and l p,α, Ark. Mat., 11, 1973, 75–101.
Constantin, P. and Saut, J.-C., Local smoothing properties of dispersive equations, J. Amer. Math. Soc., 1, 1988, 413–439.
Dias, F., Guyenne, P., Pushkarev, A. N., et al., Wave turbulence in one-dimensional models, Physica D, 152–153, 2001, 573–619.
Dudley, J., Supercontinuum generation in photonic crystal fiber, Rev. Mod. Phys., 78, 2006, 1135–1184.
Dudley, J., Genty, G. and Eggleton, B. J., Harnessing and control of optical rogue waves in supercontinum generation, Optics Express, 16, 2008, 3644–3651.
Dysthe, H., Krogstad, E. and Müller, P., Oceanic Rogue Waves, Ann. Rev. Fluid Mech., 40, 2008, 287–310.
Ghidaglia, J.-M. and Jaffard, S., Personal communication.
Hörmander, L., Estimates for translation invariant operators in L p, Acta Math., 104, 1960, 93–140.
Kharif, C. and Pelinovsky, E., Physical mechanism of rogue wave phenomenon, European J. of Mechanics, B/Fluids, 22, 2003, 603–634.
Kharif, C., Pelinovsky, E. and Slunyaev, A., RogueWaves in the Ocean, Springer-Verlag, Berlin-Heidelberg, 2009.
Linares, F. and Scialom, M., On the smoothing properties of solutions to the modified Korteweg-de Vries equation, J. Diff. Eqs., 106, 1993, 141–154.
Miyachi, A., On some singular Fourier multipliers, J. Fac. Sci. Tokyo Section I A, 28, 1981, 267–315.
Sidi, A., Sulem, C. and Sulem, P. L., On the long time behavior of a generalized KdV equation, Acta Applicandae Mathematicae, 7, 1986, 35–47.
Solli, D. R., Ropers, C., Koonath, P. and Jalali, B., Optical rogue waves, Nature, 450, 2007, 1054–1057.
Stenflo, L. and Marklund, M., Rogue waves in the atmosphere, 2009. arXiv:0911.1654v1
Tsutsumi, Y., L 2 solutions for the nonlinear Schrödinger equation and nonlinear groups, Funkcial. Ekvac., 30, 1987, 115–125.
Voronovich, V. V., Shrira, V. I. and Thomas, G., Can bottom friction suppress “freak wave” formation, J. Fluid Mech., 604, 2008, 263–296.
Wainger, S., Special trigonometric series in k-dimensions, Mem. American Math. Soc., 59, 1965.
Whitham, G. B., Linear and Nonlinear Waves, John Wiley & Sons, New York, 1974.
Zakharov, V. E., Stability of periodic waves of finite amplitude on the surface of a deep fluid, J. Appl. Mech. Tech. Phys., 9, 1968, 86–94.
Dedicated to Professor Roger Temam with Friendship and Admiration
Project supported by the Agence Nationale de la Recherche, France (No. ANR-07-BLAN-0250), the University of Illinois at Chicago, the Wolfgang Pauli Institute in Vienna, the University of Illinois at Chicago and the Université de Paris 11.
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Bona, J.L., Saut, J. Dispersive blow-up II. Schrödinger-type equations, optical and oceanic rogue waves. Chin. Ann. Math. Ser. B 31, 793–818 (2010). https://doi.org/10.1007/s11401-010-0617-0
- Rogue waves
- Dispersive blow-up
- Nonlinear dispersive equations
- Nonlinear Schrödinger equation
- Water wave equations
- Propagation in optical cables
- Weak turbulence models
2000 MR Subject Classification