Breathers and ‘black’ rogue waves of coupled nonlinear Schrödinger equations with dispersion and nonlinearity of opposite signs

• Published in: Communications in nonlinear science & numerical simulation Vol. 28; no. 1-3; pp. 28 - 38
• Main Authors: Li, Jin Hua, Chan, Hiu Ning, Chiang, Kin Seng, Chow, Kwok Wing
• Format: Journal Article
• Language: English
• Published: Elsevier B.V 01.11.2015
• Subjects: Breathers; Dispersions; Instability; Manakov systems; Mathematical analysis; Mathematical models; Modulation; Nonlinearity; Rogue waves; Schroedinger equation; Manakov systems; Rogue waves; Breathers
• ISSN: 1007-5704; 1878-7274
• DOI: 10.1016/j.cnsns.2015.03.019

•Rogue waves of coupled nonlinear Schrödinger equations are studied by the Hirota method.•Existence criterion of rogue waves correlates with that of modulation instability.•`Black' rogue waves with intensity dropping to zero are identified.•Peak displacements of rogue waves differ from those in the single waveguide case.•Numerical simulations on the evolution of breathers and rogue waves are performed. Breathers and rogue waves of special coupled nonlinear Schrödinger systems (the Manakov equations) are studied analytically. These systems model the orthogonal polarization modes in an optical fiber with randomly varying birefringence. Studies earlier in the literature had shown that rogue waves can occur in these Manakov systems with dispersion and nonlinearity of opposite signs, and that the criterion for the existence of rogue waves correlates closely with the onset of modulation instability. In the present work the Hirota bilinear transform is employed to calculate the breathers (pulsating modes), and rogue waves are obtained as a long wave limit of such breathers. In terms of wave profiles, a ‘black’ rogue wave (intensity dropping to zero) and the transition to a four-petal configuration are elucidated analytically. Sufficiently strong modulation instabilities of the background may overwhelm or mask the development of the rogue waves, and such thresholds are correlated to actual physical properties of optical fibers. Numerical simulations on the evolution of breathers are performed to verify the prediction of the analytical formulations.