Origin of 'déjà vu' phenomenon in the framework of the Fermi–Pasta–Ulam problem in nonautonomous systems

The induced modulational instability is studied in the framework of the nonautonomous nonlinear Schrodinger equation (NLSE) model with varying dispersion and nonlinearity. The Hasegawa algorithm of the analysis of the modulational instability (MI) is generalized taking into account both varying dispersion and nonlinearity. Main features of the induced MI are revealed using direct computational experiments. In particular, the induced modulational instability is considered from a point of view of dynamical memory behavior, and the main restrictions leading to imperfect restoration of the information signal are found. Our direct computer experiments offer a clearer view of how nonlinear analogies to the “deja vu” phenomena have arisen. It is shown that in the framework of the nonautonomous NLSE model, the “deja vu” phenomena must be accompanied by the imperfect spectral energy restoration into its fundamental Fourier mode, the variation of the Fermi–Pasta–Ulam recurrence period along the propagation distance, the excitation of the higher-order modulation instability, and the rogue waves formation.