From a breather homoclinic wave to a rogue wave solution for the coupled Schrödinger–Boussinesq equation

In this paper, two types of multi-parameter breather homoclinic wave solutions—including breather homoclinic wave and rational homoclinic wave solutions—are obtained by using the Hirota technique and ansatz with complexity of parameter for the coupled Schrodinger–Boussinesq equation. Rogue waves in the form of the rational homoclinic solution are derived when the periods of breather homoclinic wave go to infinite. Some novel features of homoclinic wave solutions are discussed and presented. In contrast to the normal bright rogue wave structure, a structure like a four-petaled flower in temporal–spatial distribution is exhibited. Further with the change of the wave number of the plane wave, the bright and dark rogue wave structures may change into each other. The bright rogue wave structure results from the full merger of two nearby peaks, and the dark rogue wave structure results from the full merger of two nearby holes. The dark rogue wave for the uncoupled Boussinesq equation is finally obtained. Its structural properties show that it never takes on bright rogue wave features with the change of parameter. It is hoped that these results might provide us with useful information on the dynamics of the relevant fields in physics.

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

[2]  Juekuan Yang,et al.  Molecular dynamics study of DNA translocation through graphene nanopores. , 2013, Physical review. E, Statistical, nonlinear, and soft matter physics.

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

[4]  Z. Zhang,et al.  PASSIVELY MODE-LOCKED FIBER LASER USING SOA-BASED NONLINEAR OPTICAL LOOP MIRROR , 2012 .

[5]  Junkichi Satsuma,et al.  An N-Soliton Solution for the Nonlinear Schrödinger Equation Coupled to the Boussinesq Equation , 1988 .

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

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

[8]  Harald E. Krogstad,et al.  Oceanic Rogue Waves , 2008 .

[9]  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.

[10]  Emilio Molina,et al.  Summary and Discussion , 2014 .

[11]  V. G. Makhankov,et al.  On stationary solutions of the Schrödinger equation with a self-consistent potential satisfying boussinesq's equation , 1974 .

[12]  Zhenyun Qin,et al.  Rogue Waves for the Coupled Schrödinger–Boussinesq Equation and the Coupled Higgs Equation , 2012 .

[13]  Engui Fan,et al.  A series of exact solutions for coupled Higgs field equation and coupled Schrödinger–Boussinesq equation , 2009 .

[14]  Zhenya Yan.  Financial Rogue Waves , 2009, 0911.4259.

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

[16]  Junkichi Satsuma,et al.  Soliton Solutions in a Diatomic Lattice System , 1979 .

[17]  Yasuhiro Ohta,et al.  Dynamics of rogue waves in the Davey–Stewartson II equation , 2012, 1212.0152.

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

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

[20]  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.

[21]  Peilong Chen.  Dynamics of finite-symmetry and general-shaped objects under shear and shear alignment of uniaxial objects at finite temperatures. , 2013, Physical review. E, Statistical, nonlinear, and soft matter physics.

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

[23]  J. Fatome,et al.  Observation of Kuznetsov-Ma soliton dynamics in optical fibre , 2012, Scientific Reports.

[24]  Kwok Wing Chow,et al.  Rogue Wave Modes for the Long Wave–Short Wave Resonance Model , 2013 .

[25]  K. Chow,et al.  Modulation instabilities in a system of four coupled, nonlinear Schrödinger equations , 2008 .

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

[27]  P. Clarkson,et al.  Solitons, Nonlinear Evolution Equations and Inverse Scattering: References , 1991 .

[28]  V. Konotop,et al.  Rabi oscillations of matter-wave solitons in optical lattices , 2008, 0812.5076.

[29]  Hon-Wah Tam,et al.  Homoclinic Orbits for the Coupled Schrödinger–Boussinesq Equation and Coupled Higgs Equation , 2003 .

[30]  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.

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

[32]  E. Ryabov,et al.  Intramolecular vibrational redistribution: from high-resolution spectra to real-time dynamics , 2012 .

[33]  A. Roy Chowdhury,et al.  Painléve Analysis and Backlund Transformations for Coupled Generalized Schrödinger–Boussinesq Systemfn2 , 1998 .