Degasperis–Procesi equation

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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[edit]

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[edit]

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 imbedding theorem, this means in particular that the weak solutions of the Camassa–Holm equation must be continuous with respect to x.

Peaked waves in finite water depth[edit]

In 2003, a unified wave model (UWM) for progressive gravity waves in finite water depth was proposed by Liao. Based on the symmetry and the exact wave equations, the UWM admits not only all traditional smooth periodic/solitary waves but also the peaked solitary waves including the famous peaked solitary waves of Camassa–Holm equation. Thus, the UWM unifies the smooth and peaked waves in finite water depth. In other words, the peaked solitary waves are consistent with the traditional, smooth ones, and thus are as acceptable as the smooth ones.

It is found that the peaked solitary waves in finite water depth have some unusual characteristics. First of all, it has a peaked wave elevation with a discontinuous vertical velocity v at crest. Secondly, unlike the smooth waves whose horizontal velocity u decays exponentially from free surface to the bottom, the horizontal velocity u of the peaked solitary waves always increases from free surface to the bottom. Especially, different from the smooth waves whose phase speed is dependent upon wave height, the phase speed of the peaked solitary waves in finite water depth have nothing to do with the wave height! In other words, the peaked solitary waves in finite water depth are non-dispersive.

The above usual characteristics of the peaked solitary waves in finite water depth are quite different from those of the traditional, smooth waves, and thus might challenge some traditional viewpoints. Even so, they could enrich and deepen our understandings about the peaked solitary waves, the Camassa–Holm equation and the Degasperis–Procesi equation.

Gallery[edit]

 
 
 
 
 
 
 
 
 
 
 
 


Notes[edit]

  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[edit]

Further reading[edit]