Propagation of Ion Acoustic Cnoidal Wave

Higher order contributions in the reductive perturbation theory have been reexamined for the nonlinear ion acoustic wave propagation under the periodic boundary condition. The boundary condition gives rise to renormalization terms which enable us to eliminate secular contribution in the higher order terms. As a result of analysis carried up to the second order, the ion flux associated with propagation of nonlinear ion acoustic cnoidal wave is examined for various values of excitation frequencies. It has been shown explicitly that the dependence of the ion flux on the wave amplitude becomes much weaker than that of quasi-linear theory at lower frequency. This illustrates that the averaged properties such as the averaged flux in genuine nonlinear systems depend on their nonlinear behavior in very crucial way. Range of validity of so-called quasi-linear approximation appears to be restricted to very small amplitude of disturbance.