No abstract

We study the properties of energy flux in wave turbulence via the Majda–McLaughlin– Tabak (MMT) equation with a quadratic dispersion relation. One of our purposes is to resolve the inter-scale energy flux $P$ in the stationary state to elucidate its distribution and scaling with spectral level. More importantly, we perform a quartet-level decomposition of $P=\sum _\varOmega P_\varOmega$ , with each component $P_\varOmega$ representing the contribution from quartet interactions with frequency mismatch $\varOmega$ , in order to explain the properties of $P$ as well as to study the wave turbulence closure model. Our results show that the time series of $P$ closely follows a Gaussian distribution, with its standard deviation several times its mean value $\bar {P}$ . This large standard deviation is shown to result mainly from the fluctuation of the quasi-resonances, i.e. $P_{\varOmega \neq 0}$ . The scaling of spectral level with $\bar {P}$ exhibits $\bar {P}^{1/3}$ and $\bar {P}^{1/2}$ at high and low nonlinearity, consistent with the kinetic and dynamic scalings, respectively. The different scaling laws in the two regimes are explained through the dominance of quasi-resonances ( $P_{\varOmega \neq 0}$ ) and exact-resonances ( $P_{\varOmega =0}$ ) in the former and latter regimes. Finally, we investigate the wave turbulence closure model, which connects fourth-order correlators to products of pair correlators through a broadening function $f(\varOmega )$ . Our numerical data show that consistent behaviour of $f(\varOmega )$ can be observed only upon averaging over a large number of quartets, but with such $f(\varOmega )$ showing a somewhat different form from the theory.

We analyze the energy transfer for solutions to the defocusing cubic nonlinear Schrödinger (NLS) initial value problem on 2D irrational tori. Moreover we complement the analytic study with numerical experimentation. As a biproduct of our investigation we also prove that the quasi-resonant part of the NLS initial value problem we consider, in both the focusing and defocusing case, is globally well-posed for initial data of finite mass.

We consider the nonlinear Schrödinger equation with non-local derivatives in a two-dimensional periodic domain. For certain orders of derivatives, we find a new type of breather solution dominating the field evolution at low nonlinearity levels. With the increase of nonlinearity, the breathers break down, giving way to wave turbulence (or Rayleigh-Jeans) spectra. Phase-space trajectories associated with the breather solutions are found to be close to that of the linear system, revealing a connection between the breather solution and Kolmogorov-Arnold-Moser (KAM) theory.