Degasperis–Procesi equation

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

u_{t} - u_{x x t} + 2 κ u_{x} + 4 u u_{x} = 3 u_{x} u_{x x} + u u_{x x x}

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

u_{t} - u_{x x t} + 2 κ u_{x} + (b + 1) u u_{x} = b u_{x} u_{x x} + u u_{x x x},

where $κ$ and b are real parameters (b=3 for the Degasperis–Procesi equation). It was discovered by Antonio Degasperis and Michela 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.Template:Sfnm Although discovered solely because of its mathematical properties, the Degasperis–Procesi equation (with $κ > 0$ ) has later been found to play a similar role in water wave theory as the Camassa–Holm equation.Template:Sfnm

Soliton solutions

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Among the solutions of the Degasperis–Procesi equation (in the special case $κ = 0$ ) are the so-called multipeakon solutions, which are functions of the form

u (x, t) = \sum_{i = 1}^{n} m_{i} (t) e^{- | x - x_{i} (t) |}

where the functions $m_{i}$ and $x_{i}$ satisfyTemplate:Sfn

{\dot{x}}_{i} = \sum_{j = 1}^{n} m_{j} e^{- | x_{i} - x_{j} |}, {\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.Template:Sfnm

When $κ > 0$ the soliton solutions of the Degasperis–Procesi equation are smooth; they converge to peakons in the limit as $κ$ tends to zero.Template:Sfnm

Discontinuous solutions

The Degasperis–Procesi equation (with $κ = 0$ ) is formally equivalent to the (nonlocal) hyperbolic conservation law

\partial_{t} u + \partial_{x} [\frac{u^{2}}{2} + \frac{G}{2} * \frac{3 u^{2}}{2}] = 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).Template:Sfnm 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.

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Degasperis–Procesi equation

Contents

Soliton solutions

Discontinuous solutions

Notes

References

Further reading

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Degasperis–Procesi equation

Soliton solutions

Discontinuous solutions

Notes

References

Further reading

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