Controllable rogue waves on the Jacobi-periodic background for the higher-order nonlinear Schrödinger equation

Based on Darboux transformation and nonlinearization method, we construct rogue wave solutions of the higher-order nonlinear Schrödinger equation on two distinct Jacobi-periodic wave backgrounds, specifically, dnoidal and cnoidal waves. The distribution characteristics of the Lax spectrum for the Jacobi-periodic waves are shown at the \(\lambda \) $\lambda$ complex plane. The effects of dispersion parameters \(\delta _2\) ${\delta }_2$ and \(\delta _6\) ${\delta }_6$ , the elliptic modulus k, and the integral constants C and \(C_1\) $C_1$ on the dynamical behavior of rogue wave solutions under two different periodic wave backgrounds are systematically analyzed, and shown in detail through graphics. In particular, we find that the integral constant can achieve the controllable excitation of rogue waves at fixed points and the fission of higher-order rogue waves into rogue wave pairs, which exhibit some novel dynamical phenomena.