Boundary-layer features in conservative nonlinear oscillations

This paper considers (1 + 1) dimensional conservative oscillatory systems with polynomial nonlinearities in the double limit of high amplitude (x_max → ∞) and high leading power (x^2N+1, N → ∞). In this limit, hitherto not studied in the literature, such systems exhibit behavior akin to boundary layer phenomena: The characteristics of the solution are determined by a vanishingly small segment of the amplitude near ± x_max. The oscillating entity, x(t), tends to a periodic saw-tooth shape of linear segments, the velocity, x′(t), tends to a periodic step-function and the x − x′ phase-space plot tends to a rectangle. This is demonstrated by transforming x and t into scaled variables, η and θ, respectively. η(θ) is (2-π) periodic in θ and bounded (|η(θ)|≤ 1). For large x_max, the zero-order term in an expansion in powers of a small parameter that is proportional to (1/x_max) yields an excellent approximation to the solutions of the scaled equation. The boundary-layer characteristics show up in the double limit by the fact that the deviations of η(θ), η′(θ) and the η − η′ phase-space plot from the sharp asymptotic shapes occur over a narrow range in θ of O(1/N) near the turning points of the oscillations.