Nonlinear Fourier spectral signatures of rogue waves observed in Bose-Einstein condensates
Modulation instability provides an important framework for understanding rogue wave (RW) formation on continuous backgrounds. However, the formation mechanism and nonlinear spectral structures of RWs in Bose-Einstein condensate (BEC) matter-wave systems with vanishing boundary conditions remain largely unexplored. Here, we employ the nonlinear Fourier transform (NFT), based on the integrable structure of the focusing nonlinear Schr\"odinger equation and the Zakharov-Shabat scattering problem, to investigate two representative classes of first-order RWs in BEC systems. Through nonlinear spectral analysis and Darboux reconstruction, we demonstrate that both Gaussian-wave-packet-induced extreme localization events and experimentally observed Peregrine solitons are governed by the coherent dynamics of discrete soliton modes encoded in the nonlinear spectrum. For Gaussian initial states, increasing the initial width leads to an increasing number of discrete eigenvalues, resulting in a transition from fundamental solitons and bound states to Christmas-tree-like RW structures. For experimentally observed Peregrine solitons, localized perturbations reshape the discrete spectral configuration and phase evolution, enabling coherent focusing of multiple bound soliton modes. Furthermore, we reveal the spectral mechanism of higher-order RWs and propose an inverse spectral-engineering approach based on discrete-spectrum phase matching. Our results provide a nonlinear spectral perspective for understanding and controlling RW formation in matter-wave systems with vanishing boundary conditions.