Breathing dynamics of an asymmetric Gaussian beam propagating in a saturable absorbing medium

Propagation of an asymmetric Gaussian beam in a medium with saturated nonlinear refractive index is analyzed using a "collective variable approach" to solve the general nonlinear Schrödinger equation and compared with that of a symmetric beam in both lossless and lossy media. For a lossless medium, we construct a diagram which defines regions of oscillatory and diffractive propagation of an asymmetric beam and compare it with that of a symmetric beam. We detect breathing dynamics of the widths and amplitude of the asymmetric beam in the oscillatory regime of propagation and identify two different types of width and amplitude beating, Type 1 and 2, depending on the initial beam energy and saturation constant of the medium. This is in contrast to a cubic-quintic medium where only one type of beating is obtained.