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.[1] 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.[2]

Soliton solutions

Main article: Peakon

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[3]

\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.[4]

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.[5]

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).[6] 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.

Notes

  1. ↑ Degasperis & Procesi 1999; Degasperis, Holm & Hone 2002; Mikhailov & Novikov 2002; Hone & Wang 2003; Ivanov 2005
  2. ↑ Johnson 2003; Dullin, Gottwald & Holm 2004; Constantin & Lannes 2007; Ivanov 2007
  3. ↑ Degasperis, Holm & Hone 2002
  4. ↑ Lundmark & Szmigielski 2003, 2005
  5. ↑ Matsuno 2005a, 2005b
  6. ↑ Coclite & Karlsen 2006, 2007; Lundmark 2007; Escher, Liu & Yin 2007

References

Further reading

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