Degasperis–Procesi equation

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Short description: Used in hydrology

In mathematical physics, the Degasperis–Procesi equation

[math]\displaystyle{ \displaystyle u_t - u_{xxt} + 2\kappa u_x + 4u u_x = 3 u_x u_{xx} + u u_{xxx} }[/math]

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

[math]\displaystyle{ \displaystyle u_t - u_{xxt} + 2\kappa u_x + (b+1)u u_x = b u_x u_{xx} + u u_{xxx}, }[/math]

where [math]\displaystyle{ \kappa }[/math] 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}}}, {{{2}}}) Although discovered solely because of its mathematical properties, the Degasperis–Procesi equation (with [math]\displaystyle{ \kappa \gt 0 }[/math]) has later been found to play a similar role in water wave theory as the Camassa–Holm equation.({{{1}}}, {{{2}}})

Soliton solutions

Main page: Peakon

Among the solutions of the Degasperis–Procesi equation (in the special case [math]\displaystyle{ \kappa=0 }[/math]) are the so-called multipeakon solutions, which are functions of the form

[math]\displaystyle{ \displaystyle u(x,t)=\sum_{i=1}^n m_i(t) e^{-|x-x_i(t)|} }[/math]

where the functions [math]\displaystyle{ m_i }[/math] and [math]\displaystyle{ x_i }[/math] satisfy[1]

[math]\displaystyle{ \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|}. }[/math]

These ODEs can be solved explicitly in terms of elementary functions, using inverse spectral methods.({{{1}}}, {{{2}}})

When [math]\displaystyle{ \kappa \gt 0 }[/math] the soliton solutions of the Degasperis–Procesi equation are smooth; they converge to peakons in the limit as [math]\displaystyle{ \kappa }[/math] tends to zero.({{{1}}}, {{{2}}})

Discontinuous solutions

The Degasperis–Procesi equation (with [math]\displaystyle{ \kappa=0 }[/math]) is formally equivalent to the (nonlocal) hyperbolic conservation law

[math]\displaystyle{ \partial_t u + \partial_x \left[\frac{u^2}{2} + \frac{G}{2} * \frac{3 u^2}{2} \right] = 0, }[/math]

where [math]\displaystyle{ G(x) = \exp(-|x|) }[/math], 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).({{{1}}}, {{{2}}}) In contrast, the corresponding formulation of the Camassa–Holm equation contains a convolution involving both [math]\displaystyle{ u^2 }[/math] and [math]\displaystyle{ u_x^2 }[/math], which only makes sense if u lies in the Sobolev space [math]\displaystyle{ H^1 = W^{1,2} }[/math] 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, Holm & Hone 2002.

References

  • Coclite, Giuseppe Maria; Karlsen, Kenneth Hvistendahl (2006), "On the well-posedness of the Degasperis–Procesi equation", J. Funct. Anal. 233 (1): 60–91, doi:10.1016/j.jfa.2005.07.008 
  • {{Citation
|last1=Coclite 
|first1=Giuseppe Maria 
|last2=Karlsen 
|first2=Kenneth Hvistendahl 
|year=2007 
|title=On the uniqueness of discontinuous solutions to the Degasperis–Procesi equation 
|periodical=J. Differential Equations 
|volume=234 
|issue=1 
|pages=142–160 
|url=http://www.math.uio.no/~kennethk/articles/art122_journal.pdf
|doi=10.1016/j.jde.2006.11.008 
|bibcode=2007JDE...234..142C

Further reading



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