  We propose a comprehensive model describing the Kerr nonlinear dynamics of an
electric field in a cylindrical microresonator with an effective radius
variation, coupled to a radiation source. The proposed system of equations for
coupled azimuthal modes takes into account full azimuthal dispersion as well as
the influence of the radiation source on the field in the microresonator with
the coupling coefficients determined experimentally. The model appears a
powerful tool to study nonlinear effects, generation axial-azimuthal modes and
optical frequency combs. We illustrate the power of the model with optimization
of the coupling point of the light source, getting two order of magnitude
improvement for the nonlinear threshold.
