Dispersive Revival Phenomena for the Diagonalizable Three‐Component System on the Torus

In this article, we investigate dispersive revival and fractalization phenomena in diagonalizable multi‐component dispersive systems with nonlinear interactions on the torus. We establish necessary and sufficient algebraic conditions for the occurrence of periodic dispersive revival, showing that these require integer‐coefficient polynomial dispersion relations and wavenumber‐independent eigenvectors. For multi‐component systems, the latter condition is shown to be equivalent to the simultaneous diagonalizability of the coefficient matrices. Based on this principle, we provide a complete classification of such systems into three algebraic categories and derive exact revival criteria for each via characteristic‐matrix analysis, with applications to the Zakharov and Manakov systems. Finally, numerical simulations provide evidence that revival‐like behavior can persist in the nonlinear regime when the linear part satisfies the derived quantization conditions.