Generalized Darboux transformations, semirational rogue waves, and modulation instability for the three-coupled variable-coefficient nonlinear Schrödinger system in an inhomogeneous multicomponent optical fiber

Nonlinear optics plays a crucial part in the progress of laser-based technologies and optical science. In this paper, we investigate the three-coupled variable-coefficient nonlinear Schrödinger system, which describes the amplification or attenuation of the picosecond pulses in an inhomogeneous multicomponent optical fiber with different frequencies or polarizations. Based on the existing Lax pair, we construct the first-/second-order generalized Darboux transformations and obtain the second-order semirational rogue-wave solutions, which represent the slowly varying envelopes of optical modes, under a constraint among the fiber gain/loss, nonlinearity and group velocity dispersion. We obtain the influences of nonlinearity and group velocity dispersion: when the value of the nonlinearity increases, amplitudes of the second-order semirational rogue waves decrease and when the value of the group velocity dispersion increases, amplitudes of the second-order semirational rogue waves increase. Baseband modulation instability (MI) through the linear stability explanation is obtained. When the characteristic roots have the negative imaginary parts, the system appears the baseband MI. When the MI occurs, it is of baseband type. With the positive parts, however, there is no MI occurring.