Degasperis–Procesi equation


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 and b are real parameters. It was discovered by Degasperis and Procesi in a search for integrable equations similar in form to the Camassa–Holm equation, which is the other integrable equation in this family ; that those two equations are the only integrable cases has been verified using a variety of different integrability tests. Although discovered solely because of its mathematical properties, the Degasperis–Procesi equation has later been found to play a similar role in water wave theory as the Camassa–Holm equation.

Soliton solutions

Among the solutions of the Degasperis–Procesi equation are the so-called multipeakon solutions, which are functions of the form
where the functions and satisfy
These ODEs can be solved explicitly in terms of elementary functions, using inverse spectral methods.
When the soliton solutions of the Degasperis–Procesi equation are smooth; they converge to peakons in the limit as tends to zero.

Discontinuous solutions

The Degasperis–Procesi equation is formally equivalent to the hyperbolic conservation law
where, and where the star denotes convolution with respect to x.
In this formulation, it admits weak solutions with a very low degree of regularity, even discontinuous ones. In contrast, the corresponding formulation of the Camassa–Holm equation contains a convolution involving both and, which only makes sense if u lies in the Sobolev space with respect to x. 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.