An Integrable Shallow Water Equation with Linear and Nonlinear Dispersion
Abstract
We study a class of 1+1 quadratically nonlinear water wave equations that combines the linear dispersion of the Korteweg-deVries (KdV) equation with the nonlinear/nonlocal dispersion of the Camassa-Holm (CH) equation, yet still preserves integrability via the inverse scattering transform (IST) method. This IST-integrable class of equations contains both the KdV equation and the CH equation as limiting cases. It arises as the compatibility condition for a second order isospectral eigenvalue problem and a first order equation for the evolution of its eigenfunctions. This integrable equation is shown to be a shallow water wave equation derived by asymptotic expansion at one order higher approximation than KdV. We compare its traveling wave solutions to KdV solitons.
Cite
@article{arxiv.nlin/0104004,
title = {An Integrable Shallow Water Equation with Linear and Nonlinear Dispersion},
author = {Holger R. Dullin and Georg Gottwald and Darryl D. Holm},
journal= {arXiv preprint arXiv:nlin/0104004},
year = {2016}
}
Comments
4 pages, no figures