  This paper provides a complete self-consistent nonlinear theory for electron
plasma waves, within the framework of the adiabatic approximation. The theory
applies whatever the variations of the wave amplitude, provided that they are
slow enough, and it is also valid when the plasma is inhomogeneous and non
stationary. Moreover, it accounts for: (i) the geometrical jump in action
resulting from separatrix crossing; (ii) the continuous change in phase
velocity making the wave frame non-inertial; (iii) the harmonic content of the
scalar potential ; (iv) a non-zero vector potential ; (v) the transition
probabilities from one region of phase space to the other when an orbit crosses
the separatrix ; (vi) the possible change in direction of the wavenumber. The
relative importance of each of the aforementioned effects is discussed in
detail, based on the derivation of the nonlinear frequency shift. This allows
to specify how the general formalism may be simplified, depending on the value
of the wavenumber normalized to the Debye length. Specific applications of our
theory are reported on the companion paper.
