Abstract: The weakly nonlinear theory has been widely applied in the problem of hydrodynamic stability and also in other fields. However, although its application has been successful for some problems, yet, for other problems, the results obtainedhre not satisfactory, especially for problems like transition or the evolution of the vortex in the free shear flow, for which the goal of the theoretical investigation is not seeking for a steady state, but predicting an evolutional process. In this paper, we shall examine the reason for the unsuccessfulness and suggest ways for its amendment.

Key words: hydrodynamic stability, weakly nonlinear theory, resonance