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 longwave limit of a breather (a pulsating mode), and can occur in the regime of negative cubic nonlinearity if a sufficiently strong selfsteepening 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.
 Publication:

Physical Review E
 Pub Date:
 March 2014
 DOI:
 10.1103/PhysRevE.89.032914
 Bibcode:
 2014PhRvE..89c2914C
 Keywords:

 05.45.Yv;
 02.30.Jr;
 47.35.Fg;
 Solitons;
 Partial differential equations;
 Solitary waves