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In [[mathematical physics]], the '''Degasperis–Procesi equation'''
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: <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 model|exactly solvable]] equations in the following family of third-[[Order (differential equation)|order]], non-linear, [[dispersive PDE]]s:
 
:<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 system|integrable equation]]s 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.<ref>Degasperis & Procesi 1999; Degasperis, Holm & Hone 2002; Mikhailov & Novikov 2002; Hone & Wang 2003; Ivanov 2005</ref> 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.<ref>Johnson 2003; Dullin, Gottwald & Holm 2004; Constantin & Lannes 2007; Ivanov 2007</ref>
 
== Soliton solutions ==
{{main|Peakon}}
 
Among the solutions of the Degasperis–Procesi equation (in the special case <math>\kappa=0</math>) are the so-called [[peakon|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> satisfy<ref>Degasperis, Holm & Hone 2002</ref>
 
:<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 [[Ordinary differential equation|ODEs]] can be solved explicitly in terms of elementary functions, using [[Integrable system#Solitons and inverse spectral methods|inverse spectral methods]].<ref>Lundmark & Szmigielski 2003, 2005</ref>
 
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.<ref>Matsuno 2005a, 2005b</ref>
 
== Discontinuous solutions ==
 
The Degasperis–Procesi equation (with <math>\kappa=0</math>) is formally equivalent to the (nonlocal) [[Hyperbolic partial differential equation#Hyperbolic system and conservation laws|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 solution]]s with a very low degree of regularity, even discontinuous ones ([[shock wave]]s).<ref>Coclite & Karlsen 2006, 2007; Lundmark 2007; Escher, Liu & Yin 2007</ref> 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 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 ==
 
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 <math>v</math> at crest.   Secondly, unlike the smooth waves whose horizontal velocity <math>u</math> decays exponentially from free surface to the bottom,  the horizontal velocity <math>u</math> 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==
{{Gallery
|width=250
|height=200
|align=center
|File:Degasperis-Procesi equation traveling wave plot 01.gif|
|File:Degasperis-Procesi equation traveling wave plot 02.gif|
|File:Degasperis-Procesi equation traveling wave plot 03.gif|
|File:Degasperis-Procesi equation traveling wave plot 04.gif|
|File:Degasperis-Procesi equation traveling wave plot 05.gif|
|File:Degasperis-Procesi equation traveling wave plot 06.gif|
|File:Degasperis-Procesi equation traveling wave plot 07.gif|
|File:Degasperis-Procesi equation traveling wave plot 08.gif|
|File:Degasperis-Procesi equation traveling wave plot 09.gif|
|File:Degasperis-Procesi equation traveling wave plot 10.gif|
|File:Degasperis-Procesi equation traveling wave plot 11.gif|
|File:Degasperis-Procesi equation traveling wave plot 12.gif|
}}
 
 
== Notes ==
{{reflist}}
 
== References ==
{{refbegin}}
*{{Citation
| last = Coclite
| first = Giuseppe Maria
| author-link =
| last2 = Karlsen
| first2 = Kenneth Hvistendahl
| year = 2006
| title = On the well-posedness of the Degasperis–Procesi equation
| periodical = J. Funct. Anal.
| volume = 233
| issue = 1
| pages = 60–91
| url = http://www.math.uio.no/~kennethk/articles/art113_journal.pdf
| doi = 10.1016/j.jfa.2005.07.008
}}
*{{Citation
| last = Coclite
| first = 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
}}
*{{Citation
| last = Constantin
| first = Adrian
| author-link =
| last2 = Lannes
| first2 = David
| year = 2007
| title = The hydrodynamical relevance of the Camassa–Holm and Degasperis–Procesi equations
| arxiv = 0709.0905
|bibcode = 2009ArRMA.192..165C |doi = 10.1007/s00205-008-0128-2 }}
*{{Citation
| last = Degasperis
| first = Antonio
| author-link =
| last2 = Holm
| first2 = Darryl D.
| last3 = Hone
| first3 = Andrew N. W.
| year = 2002
| title = A new integrable equation with peakon solutions
| periodical = Theoret. and Math. Phys.
| volume = 133
| issue = 2
| pages = 1463–1474
| arxiv = nlin.SI/0205023
| doi = 10.1023/A:1021186408422
}}
*{{Citation
| last = Degasperis
| first = Antonio
| last2 = Procesi
| first2 = Michela
| year = 1999
| contribution = Asymptotic integrability
| contribution-url = http://web.tiscalinet.it/SPT2001/SPT98papers/degproc98.ps
| editor-last = Degasperis
| editor-first = Antonio
| editor2-last = Gaeta
| editor2-first = Giuseppe
| title = Symmetry and Perturbation Theory (Rome, 1998)
| publication-place = River Edge, NJ
| publisher = World Scientific
| pages = 23–37
}}
*{{Citation
| last = Dullin
| first = Holger R.
| last2 = Gottwald
| first2 = Georg A.
| last3 = Holm
| first3 = Darryl D.
| year = 2004
| title = On asymptotically equivalent shallow water wave equations
| periodical = Physica D
| volume = 190
| issue =
| pages = 1–14
| arxiv =nlin.PS/0307011
| doi = 10.1016/j.physd.2003.11.004
|bibcode = 2004PhyD..190....1D }}
*{{Citation
| last = Escher
| first = Joachim
| last2 = Liu
| first2 = Yue
| last3 = Yin
| first3 = Zhaoyang
| year = 2007
| title = Shock waves and blow-up phenomena for the periodic Degasperis–Procesi equation
| periodical = Indiana Univ. Math. J.
| volume = 56
| issue = 1
| pages = 87–117
| url = http://www.iumj.indiana.edu/IUMJ/ftdload.php?year=2007&volume=56&artid=3040&ext=pdf
}}
*{{Citation
| last = Hone
| first = Andrew N. W.
| year = 2003
| last2 = Wang
| first2 = Jing Ping
| title = Prolongation algebras and Hamiltonian operators for peakon equations
| periodical = Inverse Problems
| volume = 19
| issue = 1
| pages = 129–145
| url =
| doi = 10.1088/0266-5611/19/1/307
|bibcode = 2003InvPr..19..129H }}
*{{Citation
| last = Ivanov
| first = Rossen
| year = 2005
| title = On the integrability of a class of nonlinear dispersive wave equations
| periodical = J. Nonlin. Math. Phys.
| volume = 12
| issue = 4
| pages = 462–468
| url =
| doi = 10.2991/jnmp.2005.12.4.2
|bibcode = 2005JNMP...12..462R }}
*{{Citation
| last = Ivanov
| first = Rossen
| year = 2007
| title = Water waves and integrability
| periodical = Phil. Trans. R. Soc. A
| volume = 365
| issue = 1858
| pages = 2267–2280
| url =
| doi = 10.1098/rsta.2007.2007
|bibcode = 2007RSPTA.365.2267I |arxiv = 0707.1839 }}
*{{Citation
| last = Johnson
| first = Robin S.
| year = 2003
| title = The classical problem of water waves: a reservoir of integrable and nearly-integrable equations
| periodical = J. Nonlin. Math. Phys.
| volume = 10
| issue = Supplement 1
| pages = 72–92
| doi = 10.2991/jnmp.2003.10.s1.6
|bibcode = 2003JNMP...10S..72J }}
*{{Citation
| last = Lundmark
| first = Hans
| year = 2007
| title = Formation and dynamics of shock waves in the Degasperis–Procesi equation
| periodical = J. Nonlinear Sci.
| volume = 17
| issue = 3
| pages = 169–198
| url = http://www.mittag-leffler.se/preprints/0506f/info.php?id=26
| doi = 10.1007/s00332-006-0803-3
|bibcode = 2007JNS....17..169L }}
*{{Citation
| last = Lundmark
| first = Hans
| last2 = Szmigielski
| first2 = Jacek
| year = 2003
| title = Multi-peakon solutions of the Degasperis–Procesi equation
| periodical = Inverse Problems
| volume = 19
| issue = 6
| pages = 1241–1245
| arxiv = nlin.SI/0503033
| doi = 10.1088/0266-5611/19/6/001
|bibcode = 2003InvPr..19.1241L }}
*{{Citation
| last = Lundmark
| first = Hans
| last2 = Szmigielski
| first2 = Jacek
| year = 2005
| title = Degasperis–Procesi peakons and the discrete cubic string
| periodical = Internat. Math. Res. Papers
| volume = 2005
| issue = 2
| pages = 53–116
| arxiv = nlin.SI/0503036
| doi = 10.1155/IMRP.2005.53
}}
*{{Citation
| last = Matsuno
| first = Yoshimasa
| year = 2005a
| title = Multisoliton solutions of the Degasperis–Procesi equation and their peakon limit
| periodical = Inverse Problems
| volume = 21
| issue = 5
| pages = 1553–1570
| doi = 10.1088/0266-5611/21/5/004
|arxiv = nlin/0511029 |bibcode = 2005InvPr..21.1553M }}
*{{Citation
| last = Matsuno
| first = Yoshimasa
| year = 2005b
| title = The ''N''-soliton solution of the Degasperis–Procesi equation
| periodical = Inverse Problems
| volume = 21
| issue = 6
| pages = 2085–2101
| arxiv = nlin.SI/0511029
| doi = 10.1088/0266-5611/21/6/018
|bibcode = 2005InvPr..21.2085M }}
*{{Citation
| last = Mikhailov
| first = Alexander V.
| last2 = Novikov
| first2 = Vladimir S.
| year = 2002
| title = Perturbative symmetry approach
| periodical = J. Phys. A: Math. Gen.
| volume = 35
| issue = 22
| pages = 4775–4790
| arxiv = nlin.SI/0203055v1
| doi = 10.1088/0305-4470/35/22/309
|bibcode = 2002JPhA...35.4775M }}
*{{citation
| last=Liao
| first=S.J. | authorlink = Liao Shijun
| title= Do peaked solitary water waves indeed exist?
| journal=Communications in Nonlinear Science and Numerical Simulation
|year=2013
| doi=10.1016/j.cnsns.2013.09.042
}}
{{refend}}
 
== Further reading ==
{{hidden begin
|toggle = left
|bodystyle = font-size: 100%
|title =
}}
{{refbegin}}
*{{Citation
| last = Coclite
| first = Giuseppe Maria
| author-link =
| last2 = Karlsen
| first2 = Kenneth Hvistendahl
| last3 = Risebro
| first3 = Nils Henrik
| year = 2008
| title = Numerical schemes for computing discontinuous solutions of the Degasperis–Procesi equation
| periodical = IMA J. Numer. Anal.
| volume = 28
| issue = 1
| pages = 80–105
| url = http://www.math.uio.no/~kennethk/articles/art125.pdf
| issn =
| doi = 10.1093/imanum/drm003
| oclc =
| accessdate =
}}
*{{Citation
| last = Escher
| first = Joachim
| year = 2007
| title = Wave breaking and shock waves for a periodic shallow water equation
| periodical = Phil. Trans. R. Soc. A
| volume = 365
| issue = 1858
| pages = 2281–2289
| url =
| doi = 10.1098/rsta.2007.2008
|bibcode = 2007RSPTA.365.2281E }}
*{{Citation
| last = Escher
| first = Joachim
| last2 = Liu
| first2 = Yue
| last3 = Yin
| first3 = Zhaoyang
| year = 2006
| title = Global weak solutions and blow-up structure for the Degasperis–Procesi equation
| periodical = J. Funct. Anal.
| volume = 241
| issue = 2
| pages = 457–485
| doi = 10.1016/j.jfa.2006.03.022
}}
*{{Citation
| last = Escher
| first = Joachim
| last2 = Yin
| first2 = Zhaoyang
| year = 2007
| title = On the initial boundary value problems for the Degasperis–Procesi equation
| periodical = Phys. Lett. A
| volume = 368
| issue = 1–2
| pages = 69–76
| url =
| doi = 10.1016/j.physleta.2007.03.073
|bibcode = 2007PhLA..368...69E }}
*{{Citation
| last = Guha
| first = Parta
| year = 2007
| title = Euler–Poincaré formalism of (two component) Degasperis–Procesi and Holm–Staley type systems
| periodical = J. Nonlin. Math. Phys.
| volume = 14
| issue = 3
| pages = 390–421
| doi = 10.2991/jnmp.2007.14.3.8
|bibcode = 2007JNMP...14..390G }}
*{{Citation
| last = Henry
| first = David
| year = 2005
| title = Infinite propagation speed for the Degasperis–Procesi equation
| periodical = J. Math. Anal. Appl.
| volume = 311
| issue = 2
| pages = 755–759
| doi = 10.1016/j.jmaa.2005.03.001
|bibcode = 2005JMAA..311..755H }}
*{{Citation
| last = Hoel
| first = Håkon A.
| author-link =
| year = 2007
| title = A numerical scheme using multi-shockpeakons to compute solutions of the Degasperis–Procesi equation
| periodical = Electron. J. Differential Equations
| volume = 2007
| issue = 100
| pages = 1–22
| url = http://ejde.math.txstate.edu/Volumes/2007/100/hoel.pdf
| doi =
}}
*{{Citation
| last = Lenells
| first = Jonatan
| year = 2005
| title = Traveling wave solutions of the Degasperis–Procesi equation
| periodical = J. Math. Anal. Appl.
| volume = 306
| issue = 1
| pages = 72–82
| url =
| doi = 10.1016/j.jmaa.2004.11.038
|bibcode = 2005JMAA..306...72L }}
*{{Citation
| last = Lin
| first = Zhiwu
| last2 = Liu
| first2 = Yue
| year = 2008
| title = Stability of peakons for the Degasperis–Procesi equation
| periodical = Comm. Pure Appl. Math.
| volume = 62
| issue = 1
| pages = 125–146
| arxiv = 0712.2007
| doi = 10.1002/cpa.20239
}}
*{{Citation
| last = Liu
| first = Yue
| last2 = Yin
| first2 = Zhaoyang
| year = 2006
| title = Global existence and blow-up phenomena for the Degasperis–Procesi equation
| periodical = Comm. Math. Phys.
| volume = 267
| issue = 3
| pages = 801–820
| url = http://www.mittag-leffler.se/preprints/0506f/info.php?id=22
| doi = 10.1007/s00220-006-0082-5
|bibcode = 2006CMaPh.267..801L }}
*{{Citation
| last = Liu
| first = Yue
| last2 = Yin
| first2 = Zhaoyang
| year = 2007
| title = On the blow-up phenomena for the Degasperis–Procesi equation
| periodical = Internat. Math. Res. Notices
| volume = 2007
| issue =
| pages =
| doi = 10.1093/imrn/rnm117
}}
*{{Citation
| last = Mustafa
| first = Octavian G.
| year = 2005
| title = A note on the Degasperis–Procesi equation
| periodical = J. Nonlin. Math. Phys.
| volume = 12
| issue = 1
| pages = 10–14
| doi = 10.2991/jnmp.2005.12.1.2
|bibcode = 2005JNMP...12...10M }}
*{{Citation
| last = Vakhnenko
| first = Vyacheslav O.
| last2 = Parkes
| first2 = E. John
| year = 2004
| title = Periodic and solitary-wave solutions of the Degasperis–Procesi equation
| periodical = Chaos, Solitons and Fractals
| volume = 20
| issue = 5
| pages = 1059–1073
| url = http://www.maths.strath.ac.uk/~caas35/v&pCSF04.pdf
| doi = 10.1016/j.chaos.2003.09.043
|bibcode = 2004CSF....20.1059V }}
*{{Citation
| last = Yin
| first = Zhaoyang
| year = 2003a
| title = Global existence for a new periodic integrable equation
| periodical = J. Math. Anal. Appl.
| volume = 283
| issue = 1
| pages = 129–139
| url =
| doi = 10.1016/S0022-247X(03)00250-6
}}
*{{Citation
| last = Yin
| first = Zhaoyang
| year = 2003b
| title = On the Cauchy problem for an integrable equation with peakon solutions
| periodical = Illinois J. Math.
| volume = 47
| issue = 3
| pages = 649–666.
| url = http://www.math.uiuc.edu/~hildebr/ijm/fall03/final/yin.html
| doi =
}}
*{{Citation
| last = Yin
| first = Zhaoyang
| year = 2004a
| title = Global solutions to a new integrable equation with peakons
| periodical = Indiana Univ. Math. J.
| volume = 53
| issue = 4
| pages = 1189–1209
| url =
| doi = 10.1512/iumj.2004.53.2479
}}
*{{Citation
| last = Yin
| first = Zhaoyang
| year = 2004b
| title = Global weak solutions for a new periodic integrable equation with peakon solutions
| periodical = J. Funct. Anal.
| volume = 212
| issue = 1
| pages = 182–194
| url =
| doi = 10.1016/j.jfa.2003.07.010
}}
{{refend}}
{{hidden end}}
 
{{DEFAULTSORT:Degasperis-Procesi equation}}
[[Category:Mathematical physics]]
[[Category:Solitons]]
[[Category:Partial differential equations]]
[[Category:Equations of fluid dynamics]]

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