The fractional nonlinear Schrödinger equation: Soliton turbulence, modulation instability, and extreme rogue waves

In this paper, we undertake a systematic exploration of soliton turbulent phenomena and the emergence of extreme rogue waves within the framework of the one-dimensional fractional nonlinear Schrödinger (FNLS) equation, which appears in many fields, such as nonlinear optics, Bose-Einstein condensates, plasma physics, etc. By initiating simulations with a plane wave modulated by small noise, we scrutinized the universal regimes of non-stationary turbulence through various statistical indices. Our analysis elucidates a marked increase in the probability of rogue wave occurrences as the system evolves within a certain range of Lévy index α, which can be ascribed to the broadened modulation instability bandwidth. This heightened probability of extreme rogue waves is corroborated through multiple facets, including wave-action spectrum, fourth-order moments, and probability density functions. However, it is crucial to acknowledge that a decrease in α also results in a reduction in the propagation speed of solitons within the system. Consequently, only high-amplitude solitons with non-zero background are observed, and the occurrence of collisions that could generate higher-amplitude rogue waves is suppressed. This introduces an inverse competitive mechanism: while a lower α expands the bandwidth of modulation instability, it concurrently impairs the mobility of solitons. Our findings contribute to a deeper understanding of the mechanisms driving the formation of rogue waves in nonlinear fractional systems, offering valuable insights for future theoretical and experimental studies.