Analytic iteration procedure for solitons and traveling wavefronts with sources

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.

BLUES iteration applied to nonlinear ordinary differential equations for wave propagation and heat transfer

“…In this paper we have extended a useful and accurate approach for solving nonlinear ordinary DEs with sources, based on the BLUES function method introduced in [9] and first applied quantitatively in [7]. We have shown that the method can be applied to obtain useful approximations for the oscillating wave solution of the damped nonlinear oscillator and for traveling wavefront solutions of the Fisher equation.…”

“…We present a detailed comparison with an alternative method, the ADM. In the new applications we show that the convergence of the method is sometimes local instead of the global convergence illustrated in [7]. The applications until now were for problems that can be reduced to an ordinary DE with a co-moving source.…”

“…Here we recapitulate concisely the BLUES function concept [9] from the viewpoint of the efficient iteration procedure that was developed from it in [7]. One starts from a linear ordinary DE which can be written as an operator L z acting on a function U (z) and assumes that one knows the (piecewise analytic) Green function U (z) = G(z) which solves…”

“…This approach was named BLUES (Beyond Linear Use of Equation Superposition) and it was suggested that the method can be useful as a perturbation expansion, the small parameter being the ratio of the width of the source to the decay length of the tail. Subsequently, in [7] quantitative calculations were performed, demonstrating that the method entails an analytic iteration procedure which converges exponentially rapidly for a variety of problems, and which is non-perturbative. There is no need for a small parameter.…”

“…While linear DEs with sources can be treated by the theory of Green functions and the principle of superposition, nonlinear DEs violate the superposition principle and a solution cannot be constructed in this way. In this paper, following up on our Letter [7], we investigate further how the practice of Green functions can be usefully extended to nonlinear DEs with an inhomogeneous source or sink, effectively using the superposition principle beyond the linear domain.…”