An Integrable Shallow Water Equation with Linear and Nonlinear Dispersion

Details An Integrable Shallow Water Equation with Linear and Nonlinear Dispersion An Integrable Shallow Water Equation with Linear and Nonlinear Dispersion

2001-04-02

arxiv.org/abs/nlin/0104004v1

Phys. Rev. Lett., 87, 194501 (2001)

Nonlinear Sciences

Chaotic Dynamics

4 pages, no figures

Scientific paper

10.1103/PhysRevLett.87.194501

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.

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