General high-order rogue waves and their dynamics in the nonlinear Schrödinger equation

General high-order rogue waves in the nonlinear Schrödinger equation are derived by the bilinear method. These rogue waves are given in terms of determinants whose matrix elements have simple algebraic expressions. It is shown that the general N-th order rogue waves contain N−1 free irreducible comp...

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Bibliographic Details
Published in:Proceedings of the Royal Society. A, Mathematical, physical, and engineering sciences Mathematical, physical, and engineering sciences, 2012-06, Vol.468 (2142), p.1716-1740
Main Authors: Ohta, Yasuhiro, Yang, Jianke
Format: Article
Language:English
Subjects:
Algebra
Algebraic conjugates
Amplitude
Amplitudes
Arrays
Bilinear Method
Coefficients
Determinants
Dynamics
Mathematical analysis
Mathematical expressions
Nonlinear Schrödinger Equation
Nonlinearity
Polynomials
Rogue Waves
Schroedinger equation
Solitons
Waves
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Summary:General high-order rogue waves in the nonlinear Schrödinger equation are derived by the bilinear method. These rogue waves are given in terms of determinants whose matrix elements have simple algebraic expressions. It is shown that the general N-th order rogue waves contain N−1 free irreducible complex parameters. In addition, the specific rogue waves obtained by Akhmediev et al. (Akhmediev et al. 2009 Phys. Rev. E 80, 026601 (doi:10.1103/PhysRevE.80.026601)) correspond to special choices of these free parameters, and they have the highest peak amplitudes among all rogue waves of the same order. If other values of these free parameters are taken, however, these general rogue waves can exhibit other solution dynamics such as arrays of fundamental rogue waves arising at different times and spatial positions and forming interesting patterns.
ISSN:1364-5021
1471-2946
DOI:10.1098/rspa.2011.0640