Английская Википедия:Degasperis–Procesi equation

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Шаблон:Short description In mathematical physics, the Degasperis–Procesi equation

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

where <math>\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.Шаблон:Sfnm Although discovered solely because of its mathematical properties, the Degasperis–Procesi equation (with <math>\kappa > 0</math>) has later been found to play a similar role in water wave theory as the Camassa–Holm equation.Шаблон:Sfnm

Soliton solutions

Шаблон:Main

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

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

where the functions <math>m_i</math> and <math>x_i</math> satisfyШаблон:Sfn

<math>\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.Шаблон:Sfnm

When <math>\kappa > 0</math> the soliton solutions of the Degasperis–Procesi equation are smooth; they converge to peakons in the limit as <math>\kappa</math> tends to zero.Шаблон:Sfnm

Discontinuous solutions

The Degasperis–Procesi equation (with <math>\kappa=0</math>) is formally equivalent to the (nonlocal) hyperbolic conservation law

<math>

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

where <math>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).Шаблон:Sfnm In contrast, the corresponding formulation of the Camassa–Holm equation contains a convolution involving both <math>u^2</math> and <math>u_x^2</math>, which only makes sense if u lies in the Sobolev space <math>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

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References

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Further reading

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