Modulational instability and homoclinic orbit solutions in vector nonlinear Schrödinger equation

Modulational instability has been used to explain the formation of breather and rogue waves qualitatively. In this paper, we show modulational instability can be used to explain the structure of them in a quantitative way. We develop a method to derive general forms for Akhmediev breather and rogue wave solutions in a $N$-component nonlinear Schr\"odinger equations. The existence condition for each pattern is clarified clearly. Moreover, the general multi-high-order rogue wave solutions and multi-Akhmediev breather solutions for $N$-component nonlinear Schr\"odinger equations are constructed. The results further deepen our understanding on the quantitative relations between modulational instability and homoclinic orbits solutions.

[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]  R. Hirota.  Exact envelope‐soliton solutions of a nonlinear wave equation , 1973 .

[3]  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.

[4]  C. Hamner,et al.  Dark-dark solitons and modulational instability in miscible two-component Bose-Einstein condensates , 2010, 1007.4947.

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

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

[7]  J. Koenderink.  Q… , 2014, Les noms officiels des communes de Wallonie, de Bruxelles-Capitale et de la communaute germanophone.

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

[9]  Kin Seng Chiang,et al.  Breathers and 'black' rogue waves of coupled nonlinear Schrödinger equations with dispersion and nonlinearity of opposite signs , 2015, Commun. Nonlinear Sci. Numer. Simul..

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

[11]  David W. McLaughlin,et al.  Geometry of the modulational instability III. Homoclinic orbits for the periodic sine-Gordon equation , 1990 .

[12]  Vladimir V. Konotop,et al.  Vector rogue waves in binary mixtures of Bose-Einstein condensates , 2010 .

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

[14]  A. Degasperis,et al.  Rogue waves of the vector nonlinear Schrödinger equations , 2013, 2013 Conference on Lasers & Electro-Optics Europe & International Quantum Electronics Conference CLEO EUROPE/IQEC.

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

[16]  P. Clarkson,et al.  Rational solutions of the Boussinesq equation and applications to rogue waves , 2016, 1609.00503.

[17]  Zixiang Zhou,et al.  Darboux Transformations in Integrable Systems: Theory and their Applications to Geometry , 2005 .

[18]  Umberto Bortolozzo,et al.  Rogue waves and their generating mechanisms in different physical contexts , 2013 .

[19]  Wei-Guo Zhang,et al.  Multi-rogue waves and rational solutions of the coupled nonlinear Schrödinger equations , 2013 .

[20]  Gino Biondini,et al.  Universal Nature of the Nonlinear Stage of Modulational Instability. , 2015, Physical review letters.

[21]  Liming Ling,et al.  Quantitative relations between modulational instability and several well-known nonlinear excitations , 2014, 1410.7536.

[22]  Cyril Billet,et al.  Emergent rogue wave structures and statistics in spontaneous modulation instability , 2015, Scientific Reports.

[23]  Mark Shtaif,et al.  Stokes-space analysis of modal dispersion in fibers with multiple mode transmission. , 2012, Optics express.

[24]  A. Degasperis,et al.  Integrability in Action: Solitons, Instability and Rogue Waves , 2016 .

[25]  Boling Guo,et al.  High-order rogue wave solutions for the coupled nonlinear Schrödinger equations-II , 2015, 1505.04491.

[26]  Ricardo Carretero-González,et al.  Emergent nonlinear phenomena in Bose-Einstein condensates : theory and experiment , 2008 .

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

[28]  M Lakshmanan,et al.  Exact soliton solutions, shape changing collisions, and partially coherent solitons in coupled nonlinear Schrödinger equations. , 2001, Physical review letters.

[29]  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.

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

[31]  E. Demler,et al.  Nematic order by disorder in spin-2 Bose-Einstein condensates. , 2007, Physical review letters.

[32]  Xin Wang,et al.  Periodic and rational solutions of the reduced Maxwell-Bloch equations , 2017, Commun. Nonlinear Sci. Numer. Simul..

[33]  K. Khusnutdinova.  Nonlinear waves in integrable and nonintegrable systems (Mathematical Modeling and Computation 16) By Jianke Yang , 2015 .

[34]  Govind P. Agrawal,et al.  Nonlinear Fiber Optics , 1989 .

[35]  Breathers and solitons of generalized nonlinear Schr\"odinger equations as degenerations of algebro-geometric solutions , 2011, 1106.0154.

[36]  A. Craik,et al.  Wave Interactions and Fluid Flows , 1986 .

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

[38]  Bo Tian,et al.  Bright N-soliton solutions in terms of the triple Wronskian for the coupled nonlinear Schrödinger equations in optical fibers , 2010 .

[39]  K. Bongs,et al.  Oscillations and interactions of dark and dark bright solitons in Bose Einstein condensates , 2008 .

[40]  S. Yi,et al.  Magnetic properties of a spin-3 chromium condensate , 2009, 0902.4276.

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

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

[43]  Fabio Baronio,et al.  Polarization modulation instability in a Manakov fiber system , 2015 .

[44]  M Senthilvelan,et al.  Akhmediev breathers, Ma solitons, and general breathers from rogue waves: a case study in the Manakov system. , 2013, Physical review. E, Statistical, nonlinear, and soft matter physics.

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

[46]  M. Gregory Forest,et al.  Geometry and Modulation Theory for the Periodic Nonlinear Schrodinger Equation , 1986 .

[47]  Bao-Feng Feng,et al.  General N-soliton solution to a vector nonlinear Schrödinger equation , 2014 .

[48]  Pierre Gaillard,et al.  Families of quasi-rational solutions of the NLS equation and multi-rogue waves , 2011 .

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

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

[51]  C. Hamner,et al.  Multiple dark-bright solitons in atomic Bose-Einstein condensates , 2011, 1104.4359.

[52]  M. Gregory Forest,et al.  On the construction of orbits homoclinic to plane waves in integrable coupled nonlinear Schrödinger systems , 2000 .

[54]  F. Dias,et al.  Modulation instability, Akhmediev Breathers and continuous wave supercontinuum generation. , 2009, Optics express.

[55]  Zhenya Yan,et al.  Three-component nonlinear Schrödinger equations: Modulational instability, Nth-order vector rational and semi-rational rogue waves, and dynamics , 2018, Commun. Nonlinear Sci. Numer. Simul..

[56]  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.

[57]  M. Gregory Forest,et al.  Nonfocusing Instabilities in Coupled, Integrable Nonlinear Schrödinger pdes , 2000, J. Nonlinear Sci..

[58]  P. Engels,et al.  Three-Component Soliton States in Spinor F=1 Bose-Einstein Condensates. , 2017, Physical review letters.

[59]  Efim Pelinovsky,et al.  Physical Mechanisms of the Rogue Wave Phenomenon , 2003 .