Effect of phase space folding on the accuracy of the Berk–Breizman cubic equation

We assess the quantitative accuracy of the Berk-Breizman (BB) cubic wave equation that has been previously derived to describe the initial nonlinear behavior of kinetic instabilities (Berk et al 1996 Phys. Rev. Lett. 76 1256-9; Breizman et al 1997 Phys. Plasmas 4 1559-68; Berk et al 1997 Plasma Phys. Rep. 23 778-88). The fifth-order terms in the wave equation are calculated for the first time, which allow us to quantify this accuracy arbitrarily close to marginal stability. We find a relation between the breakdown of the quantitative accuracy of the cubic equation, occurring well below the instability's saturation level, and the recently observed phase space folding effect (Idouakass 2016 PhD Thesis Aix-Marseille University). This effect emerges when the resonant particle distribution function f (phi, Omega, t) (Omega is a momentum-like variable, phi is its conjugate variable, and t is the time) evolves from a (locally) single-valued function of phi, at fixed Omega, to a triple-valued one (i.e. there emerges a point in phase space where partial derivative f /partial derivative Omega = 0).