  Localized soliton-like solutions to a $(2+1)$-dimensional hydro-dynamical
evolution equation are studied numerically. The equation is so-called
Williams-Yamagata-Flierl equation, which governs geostrophic fluid in a certain
parameter range. Although the equation does not have an integrable structure in
the ordinary sense, we find there exist shape-keeping solutions with very long
life in a special background flow and an initial condition. The stability of
the localization at the fusion process of two soliton-like objects is also
investigated. As for the indicator of the long-term stability of localization,
we propose a concept of configurational entropy, which has been introduced in
analysis for non-topological solitons in field theories.
