Convergence of Petviashvili's method near periodic waves in the fractional Korteweg-de Vries equation

TL;DR

This paper analyzes the convergence issues of Petviashvili's method when approximating periodic waves in the fractional Korteweg-de Vries equation and proposes a modification to ensure convergence.

Contribution

It explains why Petviashvili's method fails for periodic waves and introduces a simple modification that guarantees convergence.

Findings

Petviashvili's method diverges near unstable eigenvalues.

A mean value shift modification ensures unconditional convergence.

Numerical illustrations confirm the effectiveness of the modification.

Abstract

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.