  In this article, we provide a general derivation of the nonlinear frequency
shift, $\delta \omega$, for a sinusoidal electron plasma wave (EPW) that varies
slowly enough for the results derived in the companion paper, on the action
distribution function, to apply. We first consider the situation when the EPW
monotonously grows and then monotonously decays in a homogeneous plasma. In
this situation, we show a hysteresis in the wave frequency, which does not
converge back to its linear value as the wave decays to small amplitudes. We
then address the derivation of $\delta \omega$ for an EPW that keeps growing in
a one-dimensional (1-D) inhomogeneous plasma. We show that, usually, the
frequency shift does not only depend on the local EPW amplitude and wavenumber.
It also depends on the whole history of the density variations, as experienced
by the wave. In a multidimensional inhomogeneous plasma, the values assumed by
$\delta \omega$ are usually different from those derived in 1-D because, due to
the transverse electron motion, one must account for the hysteresis in $\delta
\omega$ in addition to plasma inhomogeneity. Hence, unless the EPW keeps
growing in a homogeneous one-dimensional plasma, one cannot derive $\delta
\omega$ \textit{a priori} as a function of the local wave amplitude and
wavenumber. Due to the nonlocality in the action distribution function, $\delta
\omega$ depends on the whole history of the variations of the EPW amplitude and
plasma density.
