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

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.