High-order rogue wave solutions of the classical massive Thirring model equations

Abstract The nth-order solutions of the classical massive Thirring model (MTM) equations are derived by using the n-fold Darboux transformation. These solutions are expressed by the ratios of the two determinants consisted of 2n eigenfunctions under the reduction conditions. Using this method, rogue waves are constructed explicitly up to the third-order. Three patterns, i.e., fundamental, triangular and circular patterns, of the rogue waves are discussed. The parameter μ in the MTM model plays the role of the mass in the relativistic field theory while in optics it is related to the medium periodic constant, which also results in a significant rotation and a remarkable lengthening of the first-order rogue wave. These results provide new opportunities to observe rouge waves by using a combination of electromagnetically induced transparency and the Bragg scattering four-wave mixing because of large amplitudes.

[1]  Lijuan Guo,et al.  Theoretical and experimental evidence of non-symmetric doubly localized rogue waves , 2014, Proceedings of the Royal Society A: Mathematical, Physical and Engineering Sciences.

[2]  M. Shats,et al.  Capillary rogue waves. , 2010, Physical review letters.

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

[4]  Jingsong He,et al.  Few-cycle optical rogue waves: complex modified Korteweg-de Vries equation. , 2014, Physical review. E, Statistical, nonlinear, and soft matter physics.

[5]  Shuwei Xu,et al.  Rogue wave triggered at a critical frequency of a nonlinear resonant medium. , 2016, Physical review. E.

[6]  P. McClintock,et al.  Observation of an inverse energy cascade in developed acoustic turbulence in superfluid helium. , 2008, Physical review letters.

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

[8]  Benjamin J. Eggleton,et al.  Nonlinear pulse propagation in Bragg gratings , 1997 .

[9]  Jingsong He,et al.  The Darboux transformation of the Kundu–Eckhaus equation , 2015, Proceedings of the Royal Society A: Mathematical, Physical and Engineering Sciences.

[10]  Shihua Chen,et al.  Rogue waves in coupled Hirota systems , 2013 .

[11]  Jingsong He,et al.  N-order bright and dark rogue waves in a resonant erbium-doped fiber system , 2012 .

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

[13]  Antonio Degasperis,et al.  Rational solitons of wave resonant-interaction models. , 2013, Physical review. E, Statistical, nonlinear, and soft matter physics.

[14]  Boling Guo,et al.  High-order rogue waves in vector nonlinear Schrödinger equations. , 2013, Physical review. E, Statistical, nonlinear, and soft matter physics.

[15]  Miro Erkintalo,et al.  Instabilities, breathers and rogue waves in optics , 2014, Nature Photonics.

[16]  Gui Mu,et al.  Dynamics of Rogue Waves on a Multisoliton Background in a Vector Nonlinear Schrödinger Equation , 2014, SIAM J. Appl. Math..

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

[18]  Yasuhiro Ohta,et al.  General rogue waves in the focusing and defocusing Ablowitz–Ladik equations , 2014 .

[19]  Jingsong He,et al.  The n‐order rogue waves of Fokas–Lenells equation , 2012, 1211.5924.

[20]  Q. P. Liu,et al.  Nonlinear Schrödinger equation: generalized Darboux transformation and rogue wave solutions. , 2011, Physical review. E, Statistical, nonlinear, and soft matter physics.

[21]  Wei Liu,et al.  On the evolution of a rogue wave along the orthogonal direction of the (t, x)-plane , 2015, Commun. Nonlinear Sci. Numer. Simul..

[22]  Shihua Chen,et al.  Darboux transformation and dark rogue wave states arising from two-wave resonance interaction , 2014 .

[23]  Fabio Baronio,et al.  Rogue waves emerging from the resonant interaction of three waves. , 2013, Physical review letters.

[24]  Yong Chen,et al.  Rogue wave solutions of AB system , 2013, Commun. Nonlinear Sci. Numer. Simul..

[25]  Jingsong He,et al.  The higher order rogue wave solutions of the Gerdjikov–Ivanov equation , 2013, 1304.2583.

[26]  K. Porsezian,et al.  New Types of Rogue Wave in an Erbium-Doped Fibre System , 2012 .

[27]  Gene Cooperman,et al.  Self‐pulsing and chaos in distributed feedback bistable optical devices , 1982 .

[28]  E. Kuznetsov,et al.  On the complete integrability of the two-dimensional classical Thirring model , 1977 .

[29]  Li-Chen Zhao,et al.  Localized nonlinear waves in a two-mode nonlinear fiber , 2012, 1501.05720.

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

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

[32]  Yongshuai Zhang,et al.  Darboux Transformation of the Second-Type Derivative Nonlinear Schrödinger Equation , 2014, 1408.1042.

[33]  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.

[34]  Ira Didenkulova,et al.  Rogue waves in nonlinear hyperbolic systems (shallow-water framework) , 2011 .

[35]  B. Guo,et al.  Rogue Wave, Breathers and Bright-Dark-Rogue Solutions for the Coupled Schrödinger Equations , 2011 .

[36]  Krug,et al.  Bragg grating solitons. , 1996, Physical review letters.

[37]  Li-Chen Zhao,et al.  Dynamics of nonautonomous rogue waves in Bose-Einstein condensate , 2013 .

[38]  Adrian Ankiewicz,et al.  Circular rogue wave clusters. , 2011, Physical review. E, Statistical, nonlinear, and soft matter physics.

[39]  S. K. Sharma,et al.  Observation of hole Peregrine soliton in a multicomponent plasma with critical density of negative ions , 2013 .

[40]  Darboux polynomial matrices: the classical Massive Thirring Model as study case , 2014, 1411.7965.

[41]  Yong Chen,et al.  Generalized Darboux transformation and localized waves in coupled Hirota equations , 2013, 1312.3436.

[42]  Dumitru Mihalache,et al.  Vector rogue waves in the Manakov system: diversity and compossibility , 2015 .

[43]  A. Degasperis,et al.  Bragg grating rogue wave , 2015, 1501.05644.

[44]  J. Soto-Crespo,et al.  Rogue waves and rational solutions of the nonlinear Schrödinger equation. , 2009, Physical review. E, Statistical, nonlinear, and soft matter physics.

[45]  Yasuhiro Ohta,et al.  General high-order rogue waves and their dynamics in the nonlinear Schrödinger equation , 2011, Proceedings of the Royal Society A: Mathematical, Physical and Engineering Sciences.

[46]  Bao-Feng Feng,et al.  Rational solutions to two- and one-dimensional multicomponent Yajima–Oikawa systems , 2015 .

[47]  Jingsong He,et al.  Determinant representation of Darboux transformation for the AKNS system , 2006 .

[48]  Li-Chen Zhao,et al.  Rogue-wave solutions of a three-component coupled nonlinear Schrödinger equation. , 2013, Physical review. E, Statistical, nonlinear, and soft matter physics.

[49]  Joseph,et al.  Slow Bragg solitons in nonlinear periodic structures. , 1989, Physical review letters.

[50]  A. Fokas,et al.  Generating mechanism for higher-order rogue waves. , 2012, Physical review. E, Statistical, nonlinear, and soft matter physics.

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

[52]  Shihua Chen,et al.  Dark and composite rogue waves in the coupled Hirota equations , 2014 .

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

[54]  Jingsong He,et al.  Rogue waves of the Hirota and the Maxwell-Bloch equations. , 2012, Physical review. E, Statistical, nonlinear, and soft matter physics.

[55]  V. Matveev,et al.  Darboux Transformations and Solitons , 1992 .

[56]  A. Mikhailov.  Integrability of the two-dimensional Thirring model , 1976 .

[57]  Fabio Baronio,et al.  Solutions of the vector nonlinear Schrödinger equations: evidence for deterministic rogue waves. , 2012, Physical review letters.

[58]  P. Shukla,et al.  Surface plasma rogue waves , 2011 .

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

[60]  Q. P. Liu,et al.  High‐Order Solutions and Generalized Darboux Transformations of Derivative Nonlinear Schrödinger Equations , 2012, 1205.4369.

[61]  Alejandro B. Aceves,et al.  Self-induced transparency solitons in nonlinear refractive periodic media , 1989, Annual Meeting Optical Society of America.