  Petviashvili's method has been successfully used for approximating of
solitary waves in nonlinear evolution equations. It was discovered empirically
that the method may fail for approximating of periodic waves. We consider the
case study of the fractional Korteweg-de Vries equation and explain divergence
of Petviashvili's method from unstable eigenvalues of the generalized
eigenvalue problem. We also show that a simple modification of the iterative
method after the mean value shift results in the unconditional convergence of
Petviashvili's method. The results are illustrated numerically for the
classical Korteweg-de Vries and Benjamin-Ono equations.
