On a Class of Boussinesq Equations for Shallow Water Waves

摘要

The Euler's equations describing the dynamics of capillary-gravity water waves in two-dimensions are considered in the limits of small-amplitude and long-wavelength under appropriate boundary conditions. Using a double-series perturbation analysis, a general Boussinesq type of equation is derived involving the small-amplitude and long-wavelength parameters. A recently introduced sixth-order Boussinesq equation by Daripa and Hua [Appl. Math. Comput. 101 (1999), 159-207] is recovered from this equation in the 1/3 Bond number limit (from below) when the above parameters bear a certain relationship as they approach zero.