Modified Ji-Huan He’s Frequency Formulation for Large-Amplitude Electron Plasma Oscillations

This paper examines oscillations governed by the generic nonlinear differential equation u″=ωp021−u∓2β2uγ, where ωp0, β and γ are positive constants. The aforementioned differential equation is of particular importance, as it describes electron plasma oscillations influenced by temperature effects and large oscillation amplitudes. Since no analytical solution exists for the oscillation period in terms of ωp0, β,γ and the oscillation amplitude, accurate approximations are derived. A modified He’s approach is used to account for the non-symmetrical oscillation around the equilibrium position. The motion is divided into two parts: umin≤u<ueq and ueq<u≤umax, where umin and umax are the minimum and maximum values of u, and ueq is its equilibrium value. The time intervals for each part are calculated and summed to find the oscillation period. The proposed method shows remarkable accuracy compared to numerical results. The most significant result of this paper is that He’s approach can be readily extended to strongly non-symmetrical nonlinear oscillations. It is also demonstrated that the same approach can be extended to any case where each segment of the function f(u) in the differential equation u″+fu=0 (for umin≤u<ueq and for ueq<u≤umax) can be approximated by a fifth-degree polynomial containing only odd powers.