J. Chen and D.E. Pelinovsky
Rogue periodic waves in the focusing nonlinear Schrodinger equation
Proceeding of Royal Society of London A 474 (2018), 20170814 (18 pages)
Abstract:
Rogue periodic waves stand for rogue waves on a periodic background.
The nonlinear Schrodinger equation in the focusing case admits two
families of periodic wave solutions expressed by the Jacobian elliptic
functions dn and cn. Both periodic waves are modulationally unstable with
respect to longwave perturbations. Exact solutions for the rogue periodic
waves are constructed by using the explicit expressions for the periodic
eigenfunctions of the Zakharov–Shabat spectral problem and the Darboux
transformations. These exact solutions generalize the classical rogue wave
(the so-called Peregrine’s breather). The magnification factor of the rogue
periodic waves is computed as a function of the elliptic modulus. Rogue
periodic waves constructed here are compared with the rogue wave patterns
obtained numerically in recent publications.
Keywords:
nonlinear Schrodinger equation, periodic standing waves, rogue waves, Darboux transformations