A theory proposed about 10 years ago claimed that the jovian Great Red Spot (GRS) was a solitary wave vortex (Rossby soliton) kept stationary by counter-streaming zonal winds. We have attempted to verify this soliton theory experimentally. The jovian atmosphere is modelled by a rotating thin parabolic layer of fluid (shallow water) with a free surface in which counter-streaming (zonal) flows are excited mechanically. We have found that instability of these flows can generate a Rossby autosoliton, that is, an undamped stationary solitary vortex which is alone on the perimeter of the system. This vortex rotates around its axis in the anticyclonic sense and drifts in the opposite direction to the global rotation of the system. As the observed Rossby soliton can be considered as a physical analogue (or rather as a stationary physical model) of a natural vortex such as the GRS, the results of our experiments can be considered to support the soliton theory of the GRS. We have compared the soliton model with another physical model of the GRS based on thermoconvection in rotating deep water under the action of a transverse non-monotonic temperature gradient. A new model based on a synthesis of these ideas is urgently required.