Nonlinear modulation of TM waves in a circular waveguide

The method of multiple scales is used to derive a nonlinear Schrödinger equation for the temporal and spatial amplitude and phase modulations of TM waves in a perfectly conducting guide containing a nonlinear isotropic medium. This equation is used to show that monochromatic waves are stable if the mechanism producing the nonlinearity is an electric or magnetic polarization and unstable if the nonlinearity is due to electrostriction or magnetostriction. It is also used to determine the amplitude dependence of the cutoff frequencies.