  In this paper, the closed-form analytic solutions of two new Faraday's
standing solitary waves due to the parametric resonance of liquid in a vessel
vibrating vertically with a constant frequency are given for the first time.
Using a model based on the symmetry of wave elevation and the linearized
Boussinesq equation, we gain the closed-form wave elevations of the two kinds
of non-monotonically decaying standing solitary waves with smooth crest and the
even or odd symmetry. All of them have never been reported, to the best of our
knowledge. Besides, they can well explain some experimental phenomena. All of
these are helpful to deepen and enrich our understandings about standing
solitary waves and Faraday's wave.
