Modulational instability and rogue wave solutions for the mixed focusing–defocusing semi-discrete coherently coupled nonlinear Schrödinger system with 4 × 4 Lax pair

Abstract

Under consideration is the semi-discrete mixed focusing–defocusing coherently coupled nonlinear Schrödinger system with the fourth-order auxiliary linear problem, which may describe some dynamic behaviors of light pulses. Firstly, the modulational instability is studied to analyze the possible generation reason of diverse localized waves from plane wave solution. Secondly, based on the known 4 × 4 Lax pair, the discrete generalized (m,N-m)-fold Darboux transformation is constructed and extended to solve this discrete system. Finally, as an application of the resulting Darboux transformation, some exact solutions with position control parameters are derived and shown graphically, in particular, we obtain some novel rogue wave and periodic wave solutions with only one peak and no valleys on vanishing background, which are different from the usual fundamental rogue wave with one peak and two valleys. These results might be useful for understanding propagation of light pulses.