Degasperis–Procesi equation

In mathematical physics, the Degasperis–Procesi equation


 * $$\displaystyle u_t - u_{xxt} + 2\kappa u_x + 4u u_x = 3 u_x u_{xx} + u u_{xxx}$$

is one of only two exactly solvable equations in the following family of third-order, non-linear, dispersive PDEs:


 * $$\displaystyle u_t - u_{xxt} + 2\kappa u_x + (b+1)u u_x = b u_x u_{xx} + u u_{xxx},$$

where $$\kappa$$ 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 $$\kappa > 0$$) 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 (in the special case $$\kappa=0$$) are the so-called multipeakon solutions, which are functions of the form


 * $$\displaystyle u(x,t)=\sum_{i=1}^n m_i(t) e^{-|x-x_i(t)|}$$

where the functions $$m_i$$ and $$x_i$$ satisfy


 * $$\dot{x}_i = \sum_{j=1}^n m_j e^{-|x_i-x_j|},\qquad \dot{m}_i = 2 m_i \sum_{j=1}^n m_j\, \sgn{(x_i-x_j)} e^{-|x_i-x_j|}.$$

These ODEs can be solved explicitly in terms of elementary functions, using inverse spectral methods.

When $$\kappa > 0$$ the soliton solutions of the Degasperis–Procesi equation are smooth; they converge to peakons in the limit as $$\kappa$$ tends to zero.

Discontinuous solutions
The Degasperis–Procesi equation (with $$\kappa=0$$) is formally equivalent to the (nonlocal) hyperbolic conservation law



\partial_t u + \partial_x \left[\frac{u^2}{2} + \frac{G}{2} * \frac{3 u^2}{2} \right] = 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.