NHESSD Natural Hazards and Earth System Sciences Discussions NHESSD Nat. Hazards Earth Syst. Sci. Discuss. 2195-9269 Göttingen, Germany 10.5194/nhessd-2-897-2014 Classification of homoclinic rogue wave solutions of the nonlinear Schrödinger equation Osborne A. R. 1 Nonlinear Waves Research Corporation, Arlington, Virginia, USA 29 01 2014 2 1 897 933 Copyright: © 2014 A. R. Osborne 2014 This work is licensed under the Creative Commons Attribution 3.0 Unported License. To view a copy of this licence, visit https://creativecommons.org/licenses/by/3.0/ This article is available from https://nhess.copernicus.org/preprints/2/897/2014/nhessd-2-897-2014.html The full text article is available as a PDF file from https://nhess.copernicus.org/preprints/2/897/2014/nhessd-2-897-2014.pdf

Certain <i>homoclinic solutions</i> of the nonlinear Schrödinger (NLS) equation, with spatially periodic boundary conditions, are the most common <i>unstable wave packets</i> 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 <i>other</i> 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 <i>not classifiable as homoclinic solutions</i>. These latter are the genus-2<i>N</i> theta function solutions of the NLS equation: they are the most general unstable <i>spectral solutions</i> for periodic boundary conditions. I further describe how the homoclinic solutions of the NLS equation, for <i>N = 1</i>, can be derived directly from the theta functions in a particular limit. The solutions I address herein are actual <i>spectral components</i> in the nonlinear Fourier transform theory for the NLS equation: The periodic inverse scattering transform. <br><br> The main purpose of this paper is to discuss a broader class of rogue wave packets<sup>1</sup> 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. <br><br> <sup>1</sup>An 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.

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