A generalized Petviashvili iteration method for scalar and vector Hamiltonian equations with arbitrary form of nonlinearity

TL;DR

This paper introduces a generalized Petviashvili iteration method capable of efficiently computing solitary wave solutions for a broad class of scalar and vector Hamiltonian equations with arbitrary nonlinearities, extending the original method's applicability.

Contribution

The paper presents a systematic generalization of the Petviashvili method to handle scalar and vector Hamiltonian equations with any form of nonlinearity, maintaining computational efficiency.

Findings

Successfully extended the Petviashvili method to arbitrary nonlinearities.

Maintains similar computational effort as the original method for scalar equations.

Applicable to both scalar and vector Hamiltonian equations.

Abstract

The Petviashvili's iteration method has been known as a rapidly converging numerical algorithm for obtaining fundamental solitary wave solutions of stationary scalar nonlinear wave equations with power-law nonlinearity: \ $-Mu+u^p=0$, where $M$ is a positive definite self-adjoint operator and $p={\rm const}$. In this paper, we propose a systematic generalization of this method to both scalar and vector Hamiltonian equations with arbitrary form of nonlinearity and potential functions. For scalar equations, our generalized method requires only slightly more computational effort than the original Petviashvili method.