Direct Verification of the Kinetic Description of Wave Turbulence for Finite-Size Systems Dominated by Interactions among Groups of Six Waves

The present work considers systems whose dynamics are governed by the nonlinear interactions among groups of 6 nonlinear waves, such as those described by the unforced quintic nonlinear Schrödinger equation. Specific parameter regimes in which ensemble-averaged dynamics of such systems with finite size are accurately described by a wave kinetic equation, as used in wave turbulence theory, are theoretically predicted. In addition, the underlying reasons that the wave kinetic equation may be a poor predictor of wave dynamics outside these regimes are also discussed. These theoretical predictions are directly verified by comparing ensemble averages of solutions to the dynamical equation with corresponding solutions of the wave kinetic equation.

Figure 1

Mismatch between DQNLS and WKE as a function of $L$ for the case $p=-0.6$ at $\tau ={\tau }_{\mathrm{kin}}$, $2{\tau }_{\mathrm{kin}}$, and $3{\tau }_{\mathrm{kin}}$.

Figure 2

Comparison of averaged squared amplitudes of harmonics from simulations of DQNLS and WKE for different values of parameter $p$.

Figure 3

Solutions of DQNLS at different times for $p=-1.1$, slightly below the threshold in (10) and for $p=-0.9$, slightly above the threshold in (10). In both cases $t={\tau }_{\mathrm{kin}}$ and $L=80$.