Degasperis–Procesi Equation

In mathematical physics, the Degasperis–Procesi equation

: $\displaystyle u_t - u_ + 2\kappa u_x + 4u u_x = 3 u_x u_ + u u_$

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

:$\displaystyle u_t - u_ + 2\kappa u_x + (b+1)u u_x = b u_x u_ + u u_,$

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

:$\displaystyle u(x,t)=\sum_^n m_i(t) e^$

where the functions $m_i$ and $x_i$ satisfy

:$\dot_i = \sum_^n m_j e^,\qquad \dot_i = 2 m_i \sum_^n m_j\, \sgn e^.$

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 + \frac * \frac \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 $H^1 = W^$ 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''.

Notes

References

*
*
*
*
*
*
*
*
*
*
*
*
*
*
*
*
*
*

Further reading

*
*
*
*
*
*
*
*
*
*
*
*
*
*
*
*
*

{{DEFAULTSORT:Degasperis-Procesi equation
Mathematical physics
Solitons
Partial differential equations
Equations of fluid dynamics