Vector rogue waves for the N-coupled generalized nonlinear Schrödinger equations with cubic-quintic nonlinearity in an optical fiber

Under investigation in this paper is the N-coupled generalized nonlinear Schrödinger (N-CGNLS) equations with cubic-quintic nonlinearity, which describe the effects of quintic nonlinearity on the ultrashort optical pulse propagation in N fields of an optical fiber. Lax pair and Darboux transformation (DT) for such equations are obtained. Via the modified DT, the first- and second-order vector rogue waves are analytically presented. Through the variation of the coefficient ρ1, which is relevant to the quintic nonlinearity, time-retarded induced Raman process and nonlinear dispersion, rogue-wave rotation happens, but ρ1 has no effect on the wave peaks and cores. Two different patterns of the second-order vector rogue waves for the 3-CGNLS equations are analyzed graphically: One pattern with one peak surrounded by four tails, the other pattern including three separated equal peaks. Finally, condition for the modulation instability is given.