title: Rogue wave modes for a derivative nonlinear Schrödinger model.
abstract: Rogue waves in fluid dynamics and optical waveguides are unexpectedly large displacements from a background state, and occur in the nonlinear Schrödinger equation with positive linear dispersion in the regime of positive cubic nonlinearity. Rogue waves of a derivative nonlinear Schrödinger equation are calculated in this work as a long-wave limit of a breather (a pulsating mode), and can occur in the regime of negative cubic nonlinearity if a sufficiently strong self-steepening nonlinearity is also present. This critical magnitude is shown to be precisely the threshold for the onset of modulation instabilities of the background plane wave, providing a strong piece of evidence regarding the connection between a rogue wave and modulation instability. The maximum amplitude of the rogue wave is three times that of the background plane wave, a result identical to that of the Peregrine breather in the classical nonlinear Schrödinger equation model. This amplification ratio and the resulting spectral broadening arising from modulation instability correlate with recent experimental results of water waves. Numerical simulations in the regime of marginal stability are described.
authors.affiliation: Department of Mechanical Engineering, University of Hong Kong, Pokfulam, Hong Kong.
authors.forename: Hiu Ning
authors.initials: HN
authors.lastname: Chan
authors.affiliation: Department of Mechanical Engineering, University of Hong Kong, Pokfulam, Hong Kong.
authors.forename: Kwok Wing
authors.initials: KW
authors.lastname: Chow
authors.affiliation: Research School of Physics and Engineering, Australian National University, Canberra, Australian Capital Territory 0200, Australia.
authors.forename: David Jacob
authors.initials: DJ
authors.lastname: Kedziora
authors.affiliation: Department of Mathematical Sciences, Loughborough University, Loughborough LE11 3TU, United Kingdom.
authors.forename: Roger Hamilton James
authors.initials: RH
authors.lastname: Grimshaw
authors.affiliation: Department of Mathematics and Physics, Azusa Pacific University, Azusa, California 91702, USA.
authors.forename: Edwin
authors.initials: E
authors.lastname: Ding