Journal cover Journal topic
Natural Hazards and Earth System Sciences An interactive open-access journal of the European Geosciences Union
doi:10.5194/nhessd-2-897-2014
© Author(s) 2014. This work is distributed
under the Creative Commons Attribution 3.0 License.
Research article
29 Jan 2014
Review status
A revision of this discussion paper for further review has not been submitted.
Classification of homoclinic rogue wave solutions of the nonlinear Schrödinger equation
A. R. Osborne Nonlinear Waves Research Corporation, Arlington, Virginia, USA
Abstract. Certain homoclinic solutions of the nonlinear Schrödinger (NLS) equation, with spatially periodic boundary conditions, are the most common unstable wave packets associated with the phenomenon of oceanic rogue waves. Indeed the homoclinic solutions due to Akhmediev, Peregrine and Kuznetsov-Ma are almost exclusively used in scientific and engineering applications. Herein I investigate an infinite number of other homoclinic solutions of NLS and show that they reduce to the above three classical homoclinic solutions for particular spectral values in the periodic inverse scattering transform. Furthermore, I discuss another infinity of solutions to the NLS equation that are not classifiable as homoclinic solutions. These latter are the genus-2N theta function solutions of the NLS equation: they are the most general unstable spectral solutions for periodic boundary conditions. I further describe how the homoclinic solutions of the NLS equation, for N = 1, can be derived directly from the theta functions in a particular limit. The solutions I address herein are actual spectral components in the nonlinear Fourier transform theory for the NLS equation: The periodic inverse scattering transform.

The main purpose of this paper is to discuss a broader class of rogue wave packets1 for ship design, as defined in the Extreme Seas program. The spirit of this research came from D. Faulkner (2000) who many years ago suggested that ship design procedures, in order to take rogue waves into account, should progress beyond the use of simple sine waves.

1An overview of other work in the field of rogue waves is given elsewhere: Osborne 2010, 2012 and 2013. See the books by Olagnon and colleagues 2000, 2004 and 2008 for the Brest meetings. The books by Kharif et al. (2008) and Pelinovsky et al. (2010) are excellent references.


Citation: Osborne, A. R.: Classification of homoclinic rogue wave solutions of the nonlinear Schrödinger equation, Nat. Hazards Earth Syst. Sci. Discuss., 2, 897-933, doi:10.5194/nhessd-2-897-2014, 2014.
A. R. Osborne
Interactive discussionStatus: closed (peer review stopped)
AC: Author comment | RC: Referee comment | SC: Short comment | EC: Editor comment
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RC C338: 'classification of homoclinic rogue wavess', Constance Schober, 07 Apr 2014 Printer-friendly Version 
 
RC C376: 'Referee Comment', Vladimir Matveev, 11 Apr 2014 Printer-friendly Version Supplement 
A. R. Osborne
A. R. Osborne

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