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 according to the resonance conditions. This phase combination is called dynamical phase. Its evolution is studied for two integrable cases: a triad and a cluster formed by two connected triads, using a numerical method which is fully validated by monitoring the conserved quantities known analytically. We show that dynamical phases, usually regarded as equal to zero or constants, play a substantial role in the dynamics of the clusters. Indeed, some 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 a periodic behaviour (with period $T$) 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; to guide and improve the performance of experiments; to interpret the results of numerical simulations, etc.