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An Integrable Shallow Water Equation with Linear and Nonlinear Dispersion

Holger R. Dullin·Georg Gottwald·Darryl D. Holm
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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.

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Holger R. Dullin, Georg Gottwald, Darryl D. Holm. 2001-04-02. An Integrable Shallow Water Equation with Linear and Nonlinear Dispersion. https://doi.org/10.1103/physrevlett.87.194501

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