Camassa–Holm Equation - Relation To Waves in Shallow Water

Relation To Waves in Shallow Water

The Camassa–Holm equation can be written as the system of equations:

\begin{align} u_t + u u_x + p_x &= 0, \\ p - u_{xx} &= 2 \kappa u + u^2 + \frac{1}{2} \left( u_x \right)^2, \end{align}

with p the (dimensionless) pressure or surface elevation. This shows that the Camassa–Holm equation is a model for shallow water waves with non-hydrostatic pressure and a water layer on a horizontal bed.

The linear dispersion characteristics of the Camassa–Holm equation are:

with ω the angular frequency and k the wavenumber. Not surprisingly, this is of similar form as the one for the Korteweg–de Vries equation, provided κ is unequal to zero. For κ equal to zero, the Camassa–Holm equation has no frequency dispersion — moreover, the linear phase speed is zero for this case. As a result, κ is the phase speed for the long-wave limit of k approaching zero, and the Camassa–Holm equation is (if κ is non-zero) a model for one-directional wave propagation like the Korteweg–de Vries equation.

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