Abstract
Heuristic asymptotic methods are commonly used to study the long-term development of oscillatory solutions of nonlinear ordinary differential equations and wave-type solutions of partial differential equation. For certain classes of weakly nonlinear systems, energy methods are here used to establish the validity of such approximations. There is an overall limitation of the results to a time interval determined by the time scale of significant energy transfer, but this is sufficiently long for interesting physical effects to be discussed. The basic results take the form that, when the heuristic methods yield an approximation giving a uniformly bounded small error in the differential equation, then the error in the solution is small. They do not depend strongly on the properties of the approximation, other than on simple bounds.
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Schedule a callMore From: Proceedings of the Royal Society of London. A. Mathematical and Physical Sciences
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