Superregular breathers, characteristics of nonlinear stage of modulation instability induced by higher-order effects

We study the higher-order generalized nonlinear Schrödinger (NLS) equation describing the propagation of ultrashort optical pulse in optical fibres. By using Darboux transformation, we derive the superregular breather solution that develops from a small localized perturbation. This type of solution can be used to characterize the nonlinear stage of the modulation instability (MI) of the condensate. In particular, we show some novel characteristics of the nonlinear stage of MI arising from higher-order effects: (i) coexistence of a quasi-Akhmediev breather and a multipeak soliton; (ii) two multipeak solitons propagation in opposite directions; (iii) a beating pattern followed by two multipeak solitons in the same direction. It is found that these patterns generated from a small localized perturbation do not have the analogues in the standard NLS equation. Our results enrich Zakharov’s theory of superregular breathers and could provide helpful insight on the nonlinear stage of MI in presence of the higher-order effects.

[1]  H. Washimi,et al.  Self-Trapping and Instability of Hydromagnetic Waves Along the Magnetic Field in a Cold Plasma , 1968 .

[2]  E. Kuznetsov,et al.  Solitons in a parametrically unstable plasma , 1977 .

[3]  Alwyn C. Scott,et al.  Davydov's soliton revisited , 1991 .

[4]  K. Deepamala,et al.  Davydov soliton in alpha helical proteins: higher order and discreteness effects , 1995 .

[5]  Fabio Baronio,et al.  Optical rogue waves in parametric three-wave mixing and coherent stimulated scattering , 2015 .

[6]  Wen-Li Yang,et al.  State transition induced by higher-order effects and background frequency. , 2014, Physical review. E, Statistical, nonlinear, and soft matter physics.

[7]  Lihong Wang,et al.  Breather and rogue wave solutions of a generalized nonlinear Schrödinger equation. , 2013, Physical review. E, Statistical, nonlinear, and soft matter physics.

[8]  Shihua Chen.  Twisted rogue-wave pairs in the Sasa-Satsuma equation. , 2013, Physical review. E, Statistical, nonlinear, and soft matter physics.

[9]  Lei Wang,et al.  Breather-to-soliton transitions, nonlinear wave interactions, and modulational instability in a higher-order generalized nonlinear Schrödinger equation. , 2016, Physical review. E.

[10]  B. Jalali,et al.  Optical rogue waves , 2007, Nature.

[11]  L. Kavitha,et al.  SOLITON SPIN EXCITATIONS IN AN ANISOTROPIC HEISENBERG FERROMAGNET WITH OCTUPOLE-DIPOLE INTERACTION , 1999 .

[12]  A Ankiewicz,et al.  Breather-to-soliton conversions described by the quintic equation of the nonlinear Schrödinger hierarchy. , 2015, Physical review. E, Statistical, nonlinear, and soft matter physics.

[13]  Y. Nakamura,et al.  Observation of modulational instability in a multi-component plasma with negative ions , 1993, Journal of Plasma Physics.

[14]  Yan‐Chow Ma,et al.  The Perturbed Plane‐Wave Solutions of the Cubic Schrödinger Equation , 1979 .

[15]  Lei Wang,et al.  Breather transition dynamics, Peregrine combs and walls, and modulation instability in a variable-coefficient nonlinear Schrödinger equation with higher-order effects. , 2016, Physical review. E.

[16]  Frédéric Dias,et al.  The Peregrine soliton in nonlinear fibre optics , 2010 .

[17]  Bo Tian,et al.  Conservation laws, soliton solutions and modulational instability for the higher-order dispersive nonlinear Schrödinger equation , 2009 .

[18]  K. Porsezian.  Completely integrable nonlinear Schrödinger type equations on moving space curves , 1997 .

[19]  Günter Steinmeyer,et al.  Spatiotemporal rogue events in optical multiple filamentation. , 2013, Physical review letters.

[20]  Y. Nakamura,et al.  Observation of Peregrine solitons in a multicomponent plasma with negative ions. , 2011, Physical review letters.

[21]  Laurent Larger,et al.  Optical Rogue Waves in Whispering-Gallery-Mode Resonators , 2014, 1401.0924.

[22]  Hui-Qin Hao,et al.  Breathers and multi-soliton solutions for the higher-order generalized nonlinear Schrödinger equation , 2013, Commun. Nonlinear Sci. Numer. Simul..

[23]  Wen-Li Yang,et al.  Transition, coexistence, and interaction of vector localized waves arising from higher-order effects , 2015 .

[24]  T. Davydova,et al.  Schrödinger ordinary solitons and chirped solitons: fourth-order dispersive effects and cubic-quintic nonlinearity , 2001 .

[25]  M. Lakshmanan,et al.  Effect of discreteness on the continuum limit of the Heisenberg spin chain , 1988 .

[26]  L Salasnich,et al.  Modulational instability and complex dynamics of confined matter-wave solitons. , 2003, Physical review letters.

[27]  J. Bilbault,et al.  Generation of envelope and hole solitons in an experimental transmission line , 1994 .

[28]  H. Triki,et al.  Solitary wave solutions for high dispersive cubic-quintic nonlinear Schrödinger equation , 2009 .

[29]  Hiroshi Inoue,et al.  Inverse Scattering Method for the Nonlinear Evolution Equations under Nonvanishing Conditions , 1978 .

[30]  Adrian Ankiewicz,et al.  Moving breathers and breather-to-soliton conversions for the Hirota equation , 2015, Proceedings of the Royal Society A: Mathematical, Physical and Engineering Sciences.

[31]  Fabio Baronio,et al.  Vector rogue waves and baseband modulation instability in the defocusing regime. , 2014, Physical review letters.

[32]  Rumen Iliew,et al.  Rogue waves in mode-locked fiber lasers , 2012 .

[33]  Vladimir E. Zakharov,et al.  Stability of periodic waves of finite amplitude on the surface of a deep fluid , 1968 .

[34]  Guy Millot,et al.  Collision of Akhmediev Breathers in Nonlinear Fiber Optics , 2013 .

[35]  V. Zakharov,et al.  Superregular breathers in optics and hydrodynamics: Omnipresent modulation instability beyond simple periodicity , 2015 .

[36]  N. Hoffmann,et al.  Rogue wave observation in a water wave tank. , 2011, Physical review letters.

[37]  Xin Wang,et al.  Higher-order rogue wave solutions of the Kundu–Eckhaus equation , 2014 .

[38]  U. Bandelow,et al.  Sasa-Satsuma equation: soliton on a background and its limiting cases. , 2012, Physical review. E, Statistical, nonlinear, and soft matter physics.

[39]  M. Lakshmanan,et al.  On the integrability aspects of the one-dimensional classical continuum isotropic biquadratic Heisenberg spin chain , 1992 .

[40]  C Conti,et al.  Spatial Rogue Waves in Photorefractive Ferroelectrics. , 2015, Physical review letters.

[41]  V. I. Bespalov,et al.  Filamentary Structure of Light Beams in Nonlinear Liquids , 1966 .

[42]  Alfred R. Osborne,et al.  Nonlinear Ocean Waves and the Inverse Scattering Transform , 2010 .

[43]  F. Arecchi,et al.  Non-Gaussian statistics and extreme waves in a nonlinear optical cavity. , 2009, Physical review letters.

[44]  Liming Ling,et al.  W-shaped solitons generated from a weak modulation in the Sasa-Satsuma equation. , 2015, Physical review. E.

[45]  Sarah Rothstein,et al.  Optical Solitons From Fibers To Photonic Crystals , 2016 .

[46]  S. L. Palacios,et al.  Black optical solitons for media with parabolic nonlinearity law in the presence of fourth order dispersion , 2000 .

[47]  V E Zakharov,et al.  Nonlinear stage of modulation instability. , 2012, Physical review letters.

[48]  A. A. Gelash,et al.  Superregular solitonic solutions: a novel scenario for the nonlinear stage of modulation instability , 2012, 1211.1426.

[49]  V. Shrira,et al.  What makes the Peregrine soliton so special as a prototype of freak waves? , 2010 .

[50]  G. Whitham.  A general approach to linear and non-linear dispersive waves using a Lagrangian , 1965, Journal of Fluid Mechanics.

[51]  D. H. Peregrine,et al.  Water waves, nonlinear Schrödinger equations and their solutions , 1983, The Journal of the Australian Mathematical Society. Series B. Applied Mathematics.

[52]  S. Watanabe.  Self-modulation of a nonlinear ion wave packet , 1977, Journal of Plasma Physics.

[53]  L. Ostrovsky,et al.  Modulation instability: The beginning , 2009 .

[54]  T. Brooke Benjamin,et al.  The disintegration of wave trains on deep water Part 1. Theory , 1967, Journal of Fluid Mechanics.

[55]  Thomas Brooke Benjamin,et al.  Instability of periodic wavetrains in nonlinear dispersive systems , 1967, Proceedings of the Royal Society of London. Series A. Mathematical and Physical Sciences.

[56]  E. Pelinovsky,et al.  Extreme ocean waves , 2008 .

[57]  A. Davydov,et al.  Solitary excitons in one‐dimensional molecular chains , 1973 .

[58]  M. J. Lighthill,et al.  Contributions to the Theory of Waves in Non-linear Dispersive Systems , 1965 .

[59]  V. Konotop,et al.  Matter rogue waves , 2009 .

[60]  J. Soto-Crespo,et al.  Extreme waves that appear from nowhere: On the nature of rogue waves , 2009 .

[61]  Fabio Baronio,et al.  Baseband modulation instability as the origin of rogue waves , 2015, 1502.03915.