  In a simple model of propagation of asymmetric Gaussian beams in nonlinear
waveguides, described by a reduction to ordinary differential eqautions of
generalized nonlinear Schrodinger equations (GNLSEs) with cubic-quintic (CQ)
and saturable (SAT) nonlinearities and a graded-index profile, the beam widths
exhibit two different types of beating behavior, with transitions between them.
We present an analytic model to explain these phenomena, which originate in a
1:1 resonance in a 2 degree-of-freedom Hamiltonian system. We show how small
oscillations near a fixed point close to 1:1 resonance in such a system can be
approximated using an integrable Hamiltonian and, ultimately, by a single first
order differential equation. In particular, the beating transitions can be
located from coincidences of roots of a pair of quadratic equations, with
coefficients determined (in a highly complex manner) by the internal parameters
and initial conditions of the original system. The results of the analytic
model agree with numerics of the original system over large parameter ranges,
and allow new predictions that can be verified directly. In the CQ case we
identify a band of beam energies for which there is only a single beating
transition (as opposed to 0 or 2) as the eccentricity is increased. In the SAT
case we explain the sudden (dis)appearance of beating transitions for certain
values of the other parameters as the grade-index is changed.
