Periodic modulation on phase and backgrounds for breathers and rogue waves in two-component Bose–Einstein condensates

This paper investigates the periodic modulation on phase and backgrounds for breathers and rogue waves by spin-orbit coupling (SOC) and Raman coupling in two-component Bose-Einstein condensate systems. First, linear stability analysis examines modulation instability and identifies modulation-stable and unstable parameter regimes across different parameter planes. Subsequently, we establish Lax pairs and (n,N-n)-fold generalized Darboux transformation to construct exact analytical solutions for diverse breathers and rogue waves featuring periodic phase distributions and periodic backgrounds. Based on these solutions, rigorous analysis reveals that SOC induces spatiotemporal periodic modulation on the phase and spin-density distributions of breathers and rogue waves, while Raman coupling generates periodic modulation in their backgrounds. By adjusting Raman coupling strength and initial plane-wave amplitudes and wavenumbers, backgrounds exhibiting double-periodic, single-periodic, or non-periodic modulation can be excited. Additionally, we discover higher-order breathers with curved trajectories and dual-rogue-wave structures. Nonlinear interactions among breathers and rogue waves of distinct structures and orders are systematically investigated. Finally, the stability of the analytical solutions for breathers was verified through numerical simulations. This study may facilitate a deeper understanding of periodic modulation mechanisms for localized waves under SOC and Raman coupling.