In mathematical physics, the Degasperis–Procesi equation
u t − u x x t + 2 κ u x + 4 u u x = 3 u x u x x + u u x x x is one of only two exactly solvable equations in the following family of third-order, non-linear, dispersive PDEs:
u t − u x x t + 2 κ u x + ( b + 1 ) u u x = b u x u x x + u u x x x , where κ and b are real parameters (b=3 for the Degasperis–Procesi equation). 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 (corresponding to b=2); 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 (with κ > 0 ) has later been found to play a similar role in water wave theory as the Camassa–Holm equation.
Among the solutions of the Degasperis–Procesi equation (in the special case κ = 0 ) are the so-called multipeakon solutions, which are functions of the form
u ( x , t ) = ∑ i = 1 n m i ( t ) e − | x − x i ( t ) | where the functions m i and x i satisfy
x ˙ i = ∑ j = 1 n m j e − | x i − x j | , m ˙ i = 2 m i ∑ j = 1 n m j sgn ( x i − x j ) e − | x i − x j | . These ODEs can be solved explicitly in terms of elementary functions, using inverse spectral methods.
When κ > 0 the soliton solutions of the Degasperis–Procesi equation are smooth; they converge to peakons in the limit as κ tends to zero.
The Degasperis–Procesi equation (with κ = 0 ) is formally equivalent to the (nonlocal) hyperbolic conservation law
∂ t u + ∂ x [ u 2 2 + G 2 ∗ 3 u 2 2 ] = 0 , where G ( x ) = exp ( − | x | ) , 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 (shock waves). In contrast, the corresponding formulation of the Camassa–Holm equation contains a convolution involving both u 2 and u x 2 , which only makes sense if u lies in the Sobolev space H 1 = W 1 , 2 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.
