On exact solutions of the regularized long‐wave equation: A direct approach to partially integrable equations. II. Periodic solutions

In this second of two articles (designated I and II), the bilinear transformation method is used to obtain stationary periodic solutions of the partially integrable regularized long‐wave (RLW) equation. These solutions are expressed in terms of Riemann theta functions, and this approach leads to a new and compact expression for the important dispersion relation. The periodic solution (or cnoidal wave) can be represented as an infinite sum of sech² ‘‘solitary waves’’: this remarkable property may be interpreted in the context of a nonlinear superposition principle. The RLW cnoidal wave approximates to a sinusoidal wave and a solitary wave in the limits of small and large amplitudes, respectively. Analytic approximations and error estimates are given which shed light on the character of the cnoidal wave in the different parameter regimes. Similar results are presented in brief for the related RLW Boussinesq (RLWB) equation.