On the time scales of spectral evolution of nonlinear waves

Kinetic theory for nonlinear waves predicts a kinetic time scale of spectral evolution, which is O(ε^-4 ) with ε the wave steepness. However, in (Annenkov & Shrira, 2009, PRL), a “faster” evolution on the dynamic O(ε^-2 ) time scale is identified through numerical simulations of surface gravity waves, posing a challenge to the standard kinetic equation in wave turbulence. To study this problem, we consider the evolution of nonlinear wave fields via the Majda-McLaughlin-Tabak (MMT) equation, which captures wave turbulence behavior while being free of complexities associated with surface gravity waves (e.g., the canonical transformation to remove the quadratic nonlinearity terms). For the one-dimensional MMT equation, our results show that the kinetic scaling is obtained for a considerable range of nonlinearity. Below the nonlinearity range, the kinetic scaling is prohibited in the discrete wave turbulence regime. For higher nonlinearity, dynamic scaling is observed in conjunction with the emergence of coherent structures. Results from the two-dimensional MMT equation will also be presented.