Camassa–Holm equation

In fluid dynamics, the Camassa–Holm equation is the integrable, dimensionless and non-linear partial differential equation In fluid dynamics, the Camassa–Holm equation is the integrable, dimensionless and non-linear partial differential equation The equation was introduced by Roberto Camassa and Darryl Holm as a bi-Hamiltonian model for waves in shallow water, and in this context the parameter κ is positive and the solitary wave solutions are smooth solitons. In the special case that κ is equal to zero, the Camassa–Holm equation has peakon solutions: solitons with a sharp peak, so with a discontinuity at the peak in the wave slope. The Camassa–Holm equation can be written as the system of equations: 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 non-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. Introducing the momentum m as then two compatible Hamiltonian descriptions of the Camassa–Holm equation are: