Convergence of Petviashvili's method near periodic waves in the fractional Korteweg-de Vries equation
Provides a theoretical explanation and fix for a known empirical failure of Petviashvili's method, benefiting researchers computing periodic waves in nonlinear evolution equations.
Petviashvili's method fails for periodic waves in the fractional Korteweg-de Vries equation due to unstable eigenvalues; a mean value shift modification ensures unconditional convergence, demonstrated numerically for KdV and Benjamin-Ono equations.
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.