Abstract: There are three main problems in the weakly nonlinear theory of hydrody-namic stability: (1) The radius of convergence with respect to the perturbation parameter is too small and there is no concrete estimation for it. (2) The solution has a special structure, thus in general, it can not satisfy the initial condition posed by many practical problems. (3) When the linear part of its solution does not correspond to a neutral case, there are more than one way in determining the Landau constants, and actually n...