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Periodic solutions of the intermediate long-wave equation: a nonlinear superposition principle

Lookup NU author(s): Dr Allen Parker

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Abstract

Periodic stationary-wave solutions of the intermediate long-wave (ILW) equation are derived using the bilinear transformation method, and a new expression for the dispersion relation is obtained. The class of physically important real-valued solutions is identified. These solutions may be represented as an infinite superposition of solitary-wave profiles, a property shared by the related Korteweg-de Vries (KdV) and Benjamin-Ono (BO) equation. This nonlinear superposition principle, which has been the subject of various interpretations in the literature, is discussed. The ILW periodic solution approximates to a sinusoidal wave and a solitary wave in the limits of small and large amplitudes, respectively. For intermediate amplitudes the solution can be well approximated by either a sine wave or solitary wave. In the shallow-water (KdV) limit the ILW periodic solution leads to the familiar cnoidal wave, whereas the deep-water (BO) limit yields Benjamin's periodic wave. A previously unknown expression for the cnoidal-wave dispersion relation in terms of theta functions is obtained.

Publication metadata

Author(s): Parker A

Publication type: Article

Publication status: Published

Journal: Journal of Physics A: Mathematical and General

Year: 1992

Volume: 25

Issue: 7

Pages: 2005-2032

ISSN (print): 0305-4470

ISSN (electronic): 1751-8121

Publisher: Institute of Physics Publishing Ltd.

URL: http://dx.doi.org/10.1088/0305-4470/25/7/038

DOI: 10.1088/0305-4470/25/7/038

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