Degasperis–Procesi equation

In mathematical physics, the 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 κ {displaystyle kappa } 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 {displaystyle kappa >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 {displaystyle kappa =0} ) are the so-called multipeakon solutions, which are functions of the form where the functions m i {displaystyle m_{i}} and x i {displaystyle x_{i}} satisfy These ODEs can be solved explicitly in terms of elementary functions, using inverse spectral methods. When κ > 0 {displaystyle kappa >0} the soliton solutions of the Degasperis–Procesi equation are smooth; they converge to peakons in the limit as κ {displaystyle kappa } tends to zero. The Degasperis–Procesi equation (with κ = 0 {displaystyle kappa =0} ) is formally equivalent to the (nonlocal) hyperbolic conservation law where G ( x ) = exp ⁡ ( − | x | ) {displaystyle 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 {displaystyle u^{2}} and u x 2 {displaystyle u_{x}^{2}} , which only makes sense if u lies in the Sobolev space H 1 = W 1 , 2 {displaystyle 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.