Analytic iteration procedure for solitons and traveling wavefronts with sources

TL;DR

This paper introduces an iterative analytical method based on the BLUES function approach for solving nonlinear differential equations, demonstrating rapid convergence and practical accuracy for wave solutions with sources.

Contribution

The paper presents a novel iterative procedure that converges quickly for nonlinear ODEs, extending the BLUES method to problems with source terms in physics.

Findings

Convergence is exponentially rapid towards the exact solution.

First-order approximation achieves high accuracy with simple calculations.

Method is broadly applicable to various nonlinear wave problems.

Abstract

A method is presented for calculating solutions to differential equations analytically for a variety of problems in physics. An iteration procedure based on the recently proposed BLUES (Beyond Linear Use of Equation Superposition) function method is shown to converge for nonlinear ordinary differential equations. Case studies are presented for solitary wave solutions of the Camassa-Holm equation and for traveling wavefront solutions of the Burgers equation, with source terms. The convergence of the analytical approximations towards the numerically exact solution is exponentially rapid. In practice, the zeroth-order approximation (a simple convolution) is already useful and the first-order approximation is already accurate while still easy to calculate. The type of nonlinearity can be chosen rather freely, which makes the method generally applicable.