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.

arXiv comment14 pages, 6 figures

References in corpus (5)

• An integrable shallow water equation with peaked solitons
• An appetizer to modern developments on the Kardar-Parisi-Zhang universality class
• Creation and manipulation of stable dark solitons and vortices in microcavity polariton condensates
• Arbitrarily large numbers of kink internal modes in inhomogeneous sine-Gordon equations
• BLUES function method in computational physics

Cited by in corpus (4)

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• Epidemic processes with vaccination and immunity loss studied with the BLUES function method
• The BLUES function method for second-order partial differential equations: application to a nonlinear telegrapher equation