By the derivative expansion method we solved a certain perturbation problem which well represents the phase lock loop. We derived the equation which describes the dynamics of a slowly varying parameter E, where E can be regarded as the averaged energy of the system approximately.

The phenomena of the wave propagation in a nonlinear transmission line are analyzed theoretically by the derivative expansion method. Each section of this line is constructed with a series inductor L1 and the shunt circuit consisting of a seriesed L2 - C2 element and a nonlinear capacitor C (V) in which V is a line voltage. There exist two modes; L.F. mode propagating in the frequency range 0ωωt and H.F. mode in the range ωpω, where (L2 C2)1 and (1C2/C(0)). In the strongly dispersive region the asymptotic behavior of the nonlinear waves is described by the nonlinear Schrödinger equation and by the Korteweg-de Vries equation in the weakly dispersive region. Critical frequencies ω10 and ω20 (ωpω20ω10) decompose the H.F. mode into three regions (1) ωpωω20, (2) ω20ωω10 and (3) ω10ω. The resonant interaction between the L.F. mode and the H.F. mode occurs in the frequency ω10, while the lowest order nonlinear interaction disappears in the frequency ω20. In the regions (1), (3) and the L.F. mode the plane wave is stable under the wave modulation, while the plane wave is unstable in the region (2). We derive the basic equations describing the three wave interactions and find that the plane wave with a large amplitude becomes unstable through the parametric decay instability. The wave number and maximum growth rate of the excited waves are determined.