Through solving the nonlinear Schrdinger equation that optical field in logarithmically saturable media satisfies, a set of solutions were founded, which were self-consistent multimode Hermite-Gaussian functions. Because these self-consistent solutions are much like the solutions of one-dimensional harmonic oscillator, it is assumed reasonably that the mode occupation obeys Poisson distribution, just like quantum mechanical Glauber's coherent states. The assumed Possion distribution self-consistently leads to the conclusion that there is Gaussian soliton in logarithmically saturable nonlinear media, and the relationships among the Gaussian soliton, the nonlinear coefficient and the Possion parameter are obtained. If the soliton solution exists, the nonlinear coefficient must satisfy a condition of α≥1. When α=1, there is single mode Gaussian soliton only, and the beam size must be restricted as a fixed value w=1kn0n2. Under the condition, the Gaussian beam injected in medium at waist can transmits in the nonlinear medium keeping its beam size constant, otherwise the beam size will oscillate. The oscillating form and amplitude rely directly on the input beam size and its first-order derivative which indicates the beam waist would be expanded or compressed.