Wavelet representation of plasma particle dynamics in stationary electrostatic waves

The dynamics of plasma particles in the presence of a stationary electrostatic wave is studied by expanding the motion of each particle in powers of the wave amplitude. The coefficients of this expansion are wavelets in velocity space that gradually transform into well-behaved singular functions for t ⪢ t o ; t o = ( m ∕ k 2 T ) 1 ∕ 2 . The force exerted on the plasma and the rate of kinetic energy transferred to the plasma by the wave are obtained by averaging the amplitude expansion over initial conditions. Higher-order terms show that nonlinear wave-plasma interactions are governed by a universal function M ( τ ) which is independent of the velocity distribution and depends only on τ = t ∕ t osc ; t osc = ( m ∕ k q E o ) 1 ∕ 2 . It is found that the short-time ( t ≈ t osc ) behavior of M ( τ ) describing the transition between linear and nonlinear regimes is given by M ( τ ) = 1 − ( τ 4 ∕ 2 3 4 ! ) + ( 41 τ 8 ∕ 2 7 8 ! ) . Also, using an implicit expansion of the particle motion, a global component M o ( τ ) = ( 4 ∕ τ 2 ) J 1 ( τ 2 ∕ 2 ) , 0 ⩽ τ < ∞ , of the universal function is obtained.