Effect of the dynamical phases on the nonlinear amplitudes' evolution

In this Letter we show how the nonlinear evolution of a resonant triad depends on the special combination of the modes' phases chosen accordingly to the resonance conditions. This phase combination is called dynamical phase, its evolution is studied numerically, both for a triad and for a cluster formed by two connected triads. We show that dynamical phases, usually regarded as equal to zero or constants, play substantial role in the dynamics of the clusters. Indeed, effects are (i) to diminish the period of energy exchange $\tau$ within a cluster by 20$\%$ and more; (ii) to diminish, at time scale $\tau$, the variability of wave energies by 25$\%$ and more; (iii) to generate a new time scale, $T >> \tau$, in which we observe considerable energy exchange within a cluster, as well as an increasing in the variability of modes' energies. These findings can be applied, for example, to the control of energy input, exchange and output in Tokamaks; for explanation of some experimental results, etc.