General rogue waves and modulation instability of the generalized coupled nonlinear Schrödinger system in optical pulses

We focus on rogue waves and modulation instability (MI) of the generalized coupled nonlinear Schrödinger (GCNLS) system in optical pulses. Through the Kadomtsev–Petviashvili hierarchy reduction method, general high-order rogue wave solutions in Gram determinant form at $p= p 0$ are constructed, which contain derivative operators with respect to parameters $p$ and $q$⁠. We reduce solutions to purely algebraic expressions with the aid of the elementary Schur polynomials. The multiplicity of $p 0$ determines the structures of rogue waves and generates diverse patterns. The structures of $N$th-order rogue waves are composed of $N(N+1)/2$ fundamental ones while $p 0$ is a simple root. Free parameters $a j$ play an important part in the patterns of $N$th-order rogue waves, large values of $a 3$ lead to triangle structures while large values of $a 5$ yield pentagonal shapes. When $p 0$ is a double root, rogue waves are given by $2×2$ block determinants. They are degenerate solutions with $N 1=0$ or $N 2=0$⁠, and they are non-degenerate solutions under the constraint $N 1, N 2>0$⁠. Dynamics of degenerate and non-degenerate rogue waves exhibit significant difference from the former case. MI of the GCNLS system is investigated by linear stability analysis since it is closely associated with the excitation of rogue waves. Effects of different parameters on distributions of the growth rate $G$ for MI are considered. Numerical results suggest that amplitudes $A j$ and wave numbers $k j(j=1,2)$ of the background fields control the widths and positions of MI areas. The results can help us better understand some specific physical issues, especially the propagation in optical fibers.