Degasperis–Procesi equation

In mathematical physics, the Degasperis–Procesi equation

\displaystyle u_{t}-u_{xxt}+2\kappa u_{x}+4uu_{x}=3u_{x}u_{xx}+uu_{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)uu_{x}=bu_{x}u_{xx}+uu_{xxx},

where $\kappa$ 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.^[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

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}=2m_{i}\sum _{j=1}^{n}m_{j}\,\operatorname {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 {3u^{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

References

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[FOOTNOTEDegasperisProcesi1999DegasperisHolmHone2002MikhailovNovikov2002HoneWang2003Ivanov2005-1] Degasperis & Procesi 1999; Degasperis, Holm & Hone 2002; Mikhailov & Novikov 2002; Hone & Wang 2003; Ivanov 2005.

[FOOTNOTEJohnson2003DullinGottwaldHolm2004ConstantinLannes2007Ivanov2007-2] Johnson 2003; Dullin, Gottwald & Holm 2004; Constantin & Lannes 2007; Ivanov 2007.

[FOOTNOTEDegasperisHolmHone2002-3] Degasperis, Holm & Hone 2002.

[FOOTNOTELundmarkSzmigielski2003LundmarkSzmigielski2005-4] Lundmark & Szmigielski 2003; Lundmark & Szmigielski 2005.

[FOOTNOTEMatsuno2005aMatsuno2005b-5] Matsuno 2005a; Matsuno 2005b.

[FOOTNOTECocliteKarlsen2006CocliteKarlsen2007Lundmark2007EscherLiuYin2007-6] Coclite & Karlsen 2006; Coclite & Karlsen 2007; Lundmark 2007; Escher, Liu & Yin 2007.

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[2]

[3]

[4]

[5]

[6]

Soliton solutions

Discontinuous solutions

Notes

References

Further reading