  Resonant electron interaction with whistler-mode chorus waves is recognized
as one of the main drivers of radiation belt dynamics. For moderate wave
intensity, this interaction is well described by quasi-linear theory. However,
recent statistics of parallel propagating chorus waves have demonstrated that
5-20% of the observed waves are sufficiently intense to interact nonlinearly
with electrons. Such interactions include phase trapping and phase bunching
(nonlinear scattering) effects not described by the quasi-linear diffusion. For
sufficiently long (large) wave-packets, these nonlinear effects can result in
very rapid electron acceleration and scattering. In this paper we introduce a
method to include trapping and nonlinear scattering into the kinetic equation
describing the evolution of the electron distribution function. We use
statistics of Van Allen Probes and Time History of Events and Macroscale
Interactions during Substorms (THEMIS) observations to determine the
probability distribution of intense, long wave-packets as function of power and
frequency. Then we develop an analytical model of particle resonance of an
individual particle with an intense chorus wave-packet and derive the main
properties of this interaction: probability of electron trapping, energy change
due to trapping and nonlinear scattering. These properties are combined in a
nonlocal operator acting on the electron distribution function. When multiple
waves are present, we average the obtained operator over the observed
distributions of waves and examine solutions of the resultant kinetic equation.
We also examine energy conservation and its implications in systems with the
nonlinear wave-particle interaction.
