Abstract
The focusing Nonlinear Schrödinger (NLS) equation is the simplest universal model describing the modulation instability (MI) of quasi monochromatic waves in weakly nonlinear media, considered the main physical mechanism for the appearance of rogue (anomalous) waves (RWs) in Nature. In this paper we study the numerical instabilities of the Akhmediev breather, the simplest space periodic, one-mode perturbation of the unstable background, limiting our considerations to the simplest case of one unstable mode. In agreement with recent theoretical findings of the authors, in the situation in which the round-off errors are negligible with respect to the perturbations due to the discrete scheme used in the numerical experiments, the split-step Fourier method (SSFM), the numerical output is well-described by a suitable genus 2 finite-gap solution of NLS. This solution can be written in terms of different elementary functions in different time regions and, ultimately, it shows an exact recurrence of rogue waves described, at each appearance, by the Akhmediev breather. We discover a remarkable empirical formula connecting the recurrence time with the number of time steps used in the SSFM and, via our recent theoretical findings, we establish that the SSFM opens up a vertical unstable gap whose length can be computed with high accuracy, and is proportional to the inverse of the square of the number of time steps used in the SSFM. This neat picture essentially changes when the round-off error is sufficiently large. Indeed experiments in standard double precision show serious instabilities in both the periods and phases of the recurrence. In contrast with it, as predicted by the theory, replacing the exact Akhmediev Cauchy datum by its first harmonic approximation, we only slightly modify the numerical output. Let us also remark, that the first rogue wave appearance is completely stable in all experiments and is in perfect agreement with the Akhmediev formula and with the theoretical prediction in terms of the Cauchy data.
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Notes
- 1.
The authors are very grateful to M. Sommacal for introducing us to this method and providing us with his personalized MatLab code.
References
Ablowitz, M., Herbst, B.: On homoclinic structure and numerically induced chaos for the nonlinear Schrödinger equation. SIAM J. Appl. Math. 339–351 (1990)
Ablowitz, M.J., Schober, C.M., Herbst, B.M.: Numerical chaos, roundoff errors and homoclinic manifolds. Phys. Rev. Lett. 71, 2683 (1993)
Ablowitz, M.J., Hammack, J., Henderson, D., Schober, C.M.: Long-time dynamics of the modulational instability of deep water waves. Physica D 152153, 416–433 (2001)
Akhmediev, N.N., Korneev, V.I.: Modulation instability and periodic solutions of the nonlinear Schrödinger equation. Theor. Math. Phys 69(2), 1089–1093 (1986)
Akhmediev, N.N., Eleonskii, V.M., Kulagin, N.E.: Generation of periodic sequence of picosecond pulses in an optical fibre: exact solutions. J. Exp. Theor. Phys. 61, 894–899 (1985)
Akhmediev, N.N., Eleonskii, V.M., Kulagin, N.E.: Exact first order solutions of the Nonlinear Schödinger equation. Theor. Math. Phys. 72(2), 809–818 (1987)
Akhmediev, N.N.: Nonlinear physics: Déjà vu in optics. Nature (London) 413, 267–268 (2001)
Agrawal, G.P.: Nonlinear Fiber Optics, 3rd edn. Academic Press, San Diego, USA (2001). ISBN 0-12-045143-3
Baronio, F., Degasperis, A., Conforti, M., Wabnitz, S.: Solutions of the vector nonlinear Schrödinger equations: evidence for deterministic rogue waves. Phys. Rev. Lett. 109(4), 44102 (2012)
Belokolos, E.D., Bobenko, A.I., Enolski, V.Z., Its, A.R., Matveev, V.B.: Algebro-geometric Approach in the Theory of Integrable Equations. Springer Series in Nonlinear Dynamics. Springer, Berlin (1994)
Benjamin, T.B., Feir, J.E.: The disintegration of wave trains on deep water. Part I Theory. J. Fluid Mech. 27(3), 417–430 (1967)
Biondini, G., Kovacic, G.: Inverse scattering transform for the focusing nonlinear Schrödinger equation with nonzero boundary conditions. J. Math. Phys. 55, 031506 (2014)
Biondini, G., Li, S., Mantzavinos, D.: Oscillation structure of localized perturbations in modulationally unstable media. Phys. Rev. E 94, 060201(R) (2016)
Bludov, Y.V., Konotop, V.V., Akhmediev, N.: Matter rogue waves. Phys. Rev. A 80, 033610 (2009)
Bortolozzo, U., Montina, A., Arecchi, F.T., Huignard, J.P., Residori, S.: Spatiotemporal pulses in a liquid crystal optical oscillator. Phys. Rev. Lett. 99(2), 3–6 (2007)
Calini, A., Ercolani, N.M., McLaughlin, D.W., Schober, C.M.: Mel’nikov analysis of numerically induced chaos in the nonlinear Schrödinger equation. Physica D 89, 227–260 (1996)
Calini, A., Schober, C.M.: Homoclinic chaos increases the likelihood of rogue wave formation. Phys. Lett. A 298(5–6), 335–349 (2002)
Calini, A., Schober, C.M.: Dynamical criteria for rogue waves in nonlinear Schrödinger models. Nonlinearity 25, R99–R116 (2012)
Degasperis, A., Lombardo, S.: Integrability in action: solitons, instability and rogue waves. In: Onorato M., Resitori S., Baronio F. (eds.) Rogue and Shock Waves in Nonlinear Dispersive Media. Lecture Notes in Physics. http://www.springer.com/us/book/9783319392127 (2016)
Dubard, P., Gaillard, P., Klein, C., Matveev, V.B.: On multi-rogue waves solutions of the NLS equation and positon solutions of the KdV equation. Eur. Phys. J. Spec. Top. 185, 247–258 (2010)
Dysthe, K.B., Trulsen, K.: Note on breather type solutions of the NLS as models for freak-waves. Physica Scripta. T82, 48–52 (1999)
Grinevich, P.G., Santini, P.M.: The finite gap method and the analytic description of the exact rogue wave recurrence in the periodic NLS Cauchy problem. 1. Nonlinearity, 31(11), 5258–5308 (2018)
Grinevich, P.G., Santini, P.M.: The exact rogue wave recurrence in the NLS periodic setting via matched asymptotic expansions, for 1 and 2 unstable modes. Physics Letters A. 382, 973–979 (2018)
Grinevich P.G., Santini P.M.: The finite gap method and the periodic NLS Cauchy problem of the anomalous waves, for a finite number of unstable modes. arXiv:1810.09247 (2018)
Henderson, K.L., Peregrine, D.H., Dold, J.W.: Unsteady water wave modulations: fully nonlinear solutions and comparison with the nonlinear Schrödinger equtation. Wave Motion 29, 341–361 (1999)
Hirota, R.: Direct Methods for Finding Exact Solutions of Nonlinear Evolution Equations. Lecture Notes in Mathematics, vol. 515. Springer, New York (1976)
Its, A.R., Kotljarov, V.P.: Explicit formulas for solutions of a nonlinear Schrödinger equation. Dokl. Akad. Nauk Ukrain. SSR Ser. A 1051:965–968 (1976)
Its, A.R., Rybin, A.V., Sall, M.A.: Exact integration of nonlinear Schrödinger equation. Theor. Math. Phys. 74, 20–32 (1988)
Kedziora, D.J., Ankiewicz, A., Akhmediev, N.: Second-order nonlinear Schrödinger equation breather solutions in the degenerate and rogue wave limits. Phys. Rew. E 85, 066601 (2012)
Kharif C. and Pelinovsky, E.: Physical mechanisms of the rogue wave phenomenon. Eur. J. Mech. B/ Fluids J. Mech. 22, 603–634 (2004)
Kharif, C., Pelinovsky, E.: Focusing of nonlinear wave groups in deep water. JETP Lett. 73, 170–175 (2001)
Kimmoun, O., Hsu, H.C., Branger, H., Li, M.S., Chen, Y.Y., Kharif, C., Onorato, M., Kelleher, E.J.R., Kibler, B., Akhmediev, N., Chabchoub, A.: Modulation instability and phase-shifted Fermi-Pasta-Ulam recurrence. Sci. Rep. 6, 28516 (2016)
Krichever, I.M.: Methods of algebraic Geometry in the theory on nonlinear equations. Russ. Math. Surv. 32, 185–213 (1977)
Krichever, I.M.: Spectral theory of two-dimensional periodic operators and its applications. Russ. Math. Surv. 44(2), 145–225 (1989)
Krichever, I.M.: Perturbation theory in periodic problems for two-dimensional integrable systems. Sov. Sci. Rev., Sect. C, Math. Phys. Rev. 9(2), 1–103 (1992)
Kuznetsov, E.A.: Solitons in a parametrically unstable plasma. Sov. Phys. Dokl. 22, 507–508 (1977)
Kuznetsov, E.A.: Fermi-Pasta-Ulam recurrence and modulation instability. JETP Lett. 105(2), 125–129 (2017)
Lake, B.M., Yuen, H.C., Rungaldier, H., Ferguson, W.E.: Nonlinear deep-water waves: theory and experiment. Part 2 Evolution of a continuous wave train. J. Fluid Mech. 83(2), 49–74 (1977)
Ma, Y.-C.: The perturbed plane wave solutions of the cubic Schrödinger equation. Stud. Appl. Math. 60, 43–58 (1979)
Matveev, V.B., Salle, M.A.: Darboux Transformations and Solitons. Springer Series in Nonlinear Dynamics. Springer, Berlin (1991)
Novikov, S.P.: The periodic problem for the Korteweg-de Vries equation. Funct. Anal. Appl. 8(3), 236–246 (1974)
Onorato, M., Residori, S., Bortolozzo, U., Montina, A., Arecchi, F.T.: Rogue waves and their generating mechanisms in different physical contexts. Phys. Rep. 528, 47–89 (2013)
Osborne, A., Onorato, M., Serio, M.: The nonlinear dynamics of rogue waves and holes in deep-water gravity wave trains. Phys. Lett. A 275, 386–393 (2000)
Peregrine, D.H.: Water waves, nonlinear Schrödinger equations and their solutions. J. Austral. Math. Soc. Ser. B 25, 16–43 (1983)
Pierangeli, D., Di Mei, F., Conti, C., Agranat, A.J., DelRe, E.: Spatial rogue waves in photorefractive ferroelectrics. PRL 115, 093901 (2015)
Salasnich, L., Parola, A., Reatto, L.: Modulational instability and complex dynamics of confined matter-wave solitons. Phys. Rev. Lett. 91, 080405 (2003)
Smirnov, A.O.: Periodic two-phase rogue waves. Math. Not. 94(6), 897–907 (2013)
Solli, D.R., Ropers, C., Koonath, P., Jalali, B.: Optical rogue waves. Nature 450, 1054–1057 (2007)
Stokes, G.: On the theory of oscillatory waves. In: Transactions of the Cambridge Philosophical Society, vol. VIII, 197229, and Supplement 314326 (1847)
Sulem, C., Sulem, P.-L.: The Nonlinear Schrödinger Equation (Self Focusing and Wave Collapse). Springer, Berlin (1999)
Vespalov, V.I., Talanov, V.I.: Filamentary structure of light beams in nonlinear liquids. JETP Lett. 3(12), 307 (1966)
Taniuti, T., Washimi, H.: Self-trapping and instability of hydromagnetic waves along the magnetic field in a cold plasma. Phys. Rev. Lett. 21, 209–212 (1968)
Weideman, J.A.C., Herbst, B.M.: Split-step methods for the solution of the nonlinear Schrödinger equation. SIAM J. Numer. Anal. 23, 485–507 (1986)
Taha, T.R., Xu, X.: Parallel split-step fourier methods for the coupled nonlinear Schrödinger type equations. J Supercomput. 5, 5–23 (2005)
Van Simaeys, G., Emplit, P., Haelterman, M.: Experimental demonstration of the Fermi-Pasta-Ulam recurrence in a modulationally unstable optical wave. Phys. Rev. Lett. 87, 033902 (2001)
Yuen, H.C., Ferguson, W.E.: Relationship between Benjamin-Feir instability and recurrence in the nonlinear Schrödinger equation. Phys. Fluids 21, 1275 (1978)
Yuen, H., Lake, B.: Nonlinear dynamics of deep-water gravity waves. Adv. Appl. Mech. 22, 67–229 (1982)
Zakharov, V.E.: Stability of period waves of finite amplitude on surface of a deep fluid. JAMTP 9(2), 190–194 (1968)
Zakharov, V.E., Shabat, A.B.: Exact theory of two-dimensional self-focusing and one-dimensional self-modulation of waves in nonlinear media. Sov. Phys. JETP 34(1), 62–69 (1972)
Zakharov, V.E., Shabat, A.B.: A scheme for integrating the nonlinear equations of mathematical physics by the method of the inverse scattering transform I. Funct. Anal. Appl. 8, 226–235 (1974)
Zakharov, V.E., Mikhailov, A.V.: Relativistically invariant two-dimensional models of field theory which are integrable by means of the inverse scattering problem method. Sov. Phys. JETP 47, 1017–27 (1978)
Zakharov, V.E., Gelash, A.A.: On the nonlinear stage of Modulation Instability. PRL 111, 054101 (2013)
Zakharov, V., Ostrovsky, L.: Modulation instability: the beginning. Phys. D Nonlinear Phenom. 238(5), 540–548 (2009)
Acknowledgments
Two visits of P. G. Grinevich to Roma were supported by the University of Roma “La Sapienza”, and by the INFN, Sezione di Roma. P. G. Grinevich and P. M. Santini acknowledge the warm hospitality and the local support of CIC, Cuernavaca, Mexico, in December 2016. P.G. Grinevich was also partially supported by RFBR grant 17-51-150001. We acknowledge useful discussions with F. Briscese, F. Calogero, C. Conti, E. DelRe, A. Degasperis, A. Gelash, I. Krichever, A. Its, S. Lombardo, A. Mikhailov, D. Pierangeli, M. Sommacal and V. Zakharov.
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Grinevich, P.G., Santini, P.M. (2018). Numerical Instability of the Akhmediev Breather and a Finite-Gap Model of It. In: Buchstaber, V., Konstantinou-Rizos, S., Mikhailov, A. (eds) Recent Developments in Integrable Systems and Related Topics of Mathematical Physics. MP 2016. Springer Proceedings in Mathematics & Statistics, vol 273. Springer, Cham. https://doi.org/10.1007/978-3-030-04807-5_2
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