In mathematical physics, the Degasperis–Procesi equation is one of only two exactly solvable equations in the following family of third-order, non-linear, dispersive PDEs: where
It was discovered by Antonio Degasperis and Michela Procesi in a search for integrable equations similar in form to the Camassa–Holm equation, which is the other integrable equation in this family (corresponding to b=2); that those two equations are the only integrable cases has been verified using a variety of different integrability tests.
[1] Although discovered solely because of its mathematical properties, the Degasperis–Procesi equation (with
) has later been found to play a similar role in water wave theory as the Camassa–Holm equation.
satisfy[3] These ODEs can be solved explicitly in terms of elementary functions, using inverse spectral methods.
the soliton solutions of the Degasperis–Procesi equation are smooth; they converge to peakons in the limit as
) is formally equivalent to the (nonlocal) hyperbolic conservation law where
In this formulation, it admits weak solutions with a very low degree of regularity, even discontinuous ones (shock waves).
[6] In contrast, the corresponding formulation of the Camassa–Holm equation contains a convolution involving both
, which only makes sense if u lies in the Sobolev space
By the Sobolev embedding theorem, this means in particular that the weak solutions of the Camassa–Holm equation must be continuous with respect to x.