Convergence of Petviashvili's Method near Periodic Waves in the Fractional Korteweg--de Vries Equation

Petviashvili's method has been successfully used for approximating solitary waves in nonlinear evolution equations. It was discovered empirically that the method may fail when approximating 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.