- Degasperis-Procesi equation
In
mathematical physics , the Degasperis-Procesi equation: displaystyle u_t - u_{xxt} + 2kappa 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 PDE s::displaystyle u_t - u_{xxt} + 2kappa 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. [Degasperis & Procesi 1999; Degasperis, Holm & Hone 2002; Mikhailov & Novikov 2002; Hone & Wang 2003; Ivanov 2005] 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. [Johnson 2003; Dullin, Gottwald & Holm 2004; Constantin & Lannes 2007; Ivanov 2007]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 [Degasperis, Holm & Hone 2002]
: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. [Lundmark & Szmigielski 2003, 2005]
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. [Matsuno 2005a, 2005b]
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} ight] = 0,
where G(x) = exp(-|x|), and where the star denotes
convolution with respect to "x".In this formulation, it admitsweak solution s with a very low degree of regularity, even discontinuous ones (shock waves). [Coclite & Karlsen 2006, 2007; Lundmark 2007; Escher, Liu & Yin 2007] 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 theSobolev space H^1 = W^{1,2} with respect to "x". By theSobolev imbedding theorem , this means in particular that the weak solutions of the Camassa–Holm equation must be continuous with respect to "x".Notes
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