Degasperis-Procesi equation

In mathematical physics, the Degasperis-Procesi equation

: $displaystyle\; u\_t\; -\; u\_\{xxt\}\; +\; 2kappa\; u\_x\; +\; 4u\; u\_x\; =\; 3\; u\_x\; u\_\{xx\}\; +\; u\; u\_\{xxx\}$

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

:$displaystyle\; u\_t\; -\; u\_\{xxt\}\; +\; 2kappa\; u\_x\; +\; (b+1)u\; u\_x\; =\; b\; u\_x\; u\_\{xx\}\; +\; u\; u\_\{xxx\},$

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. [Degasperis & Procesi 1999; Degasperis, Holm & Hone 2002; Mikhailov & Novikov 2002; Hone & Wang 2003; Ivanov 2005] 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. [Johnson 2003; Dullin, Gottwald & Holm 2004; Constantin & Lannes 2007; Ivanov 2007]

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 [Degasperis, Holm & Hone 2002]

:$dot\{x\}\_i\; =\; sum\_\{j=1\}^n\; m\_j\; e^\{-|x\_i-x\_j,qquad\; dot\{m\}\_i\; =\; 2\; m\_i\; sum\_\{j=1\}^n\; m\_j,\; sgn\{(x\_i-x\_j)\}\; e^\{-|x\_i-x\_j.$

These ODEs can be solved explicitly in terms of elementary functions, using inverse spectral methods. [Lundmark & Szmigielski 2003, 2005]

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. [Matsuno 2005a, 2005b]

Discontinuous solutions

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

:$partial\_t\; u\; +\; partial\_x\; left\; [frac\{u^2\}\{2\}\; +\; frac\{G\}\{2\}\; *\; frac\{3\; u^2\}\{2\}\; ight]\; =\; 0,$

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). [Coclite & Karlsen 2006, 2007; Lundmark 2007; Escher, Liu & Yin 2007] 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".

Notes

References

*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.
series =
publication-place =
place =
publisher =
volume = 233
issue = 1
pages = 60–91
url = http://www.math.uio.no/~kennethk/articles/art113_journal.pdf
issn =
doi = 10.1016/j.jfa.2005.07.008
oclc =
accessdate =

*Citation
last = Coclite
first = Giuseppe Maria
author-link =
last2 = Karlsen
first2 = Kenneth Hvistendahl
year = 2007
title = On the uniqueness of discontinuous solutions to the Degasperis-Procesi equation
periodical = J. Differential Equations
series =
publication-place =
place =
publisher =
volume = 234
issue = 1
pages = 142–160
url = http://www.math.uio.no/~kennethk/articles/art122_journal.pdf
issn =
doi = 10.1016/j.jde.2006.11.008
oclc =
accessdate =

*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 = Constantin
first = Adrian
author-link =
last2 = Lannes
first2 = David
year = 2007
title = The hydrodynamical relevance of the Camassa–Holm and Degasperis–Procesi equations
periodical = Preprint arXiv:0709.0905v1 [math.AP]
volume =
issue =
pages =
url = http://arxiv.org/abs/0709.0905
issn =
doi =
oclc =
accessdate =

*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
url = http://arxiv.org/abs/nlin.SI/0205023
doi = 10.1023/A:1021186408422

*Citation
last = Degasperis
first = Antonio
author-link =
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
editor-link =
editor2-last = Gaeta
editor2-first = Giuseppe
editor2-link =
title = Symmetry and Perturbation Theory (Rome, 1998)
publication-place = River Edge, NJ
publisher = World Scientific
pages = 23–37
isbn =
doi =
oclc =
url =

*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
url =http://arxiv.org/abs/nlin.PS/0307011
doi = 10.1016/j.physd.2003.11.004

*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 = 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 = 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

*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

*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

*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

*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 = 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

*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

*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

*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

*Citation
last = Lin
first = Zhiwu
last2 = Liu
first2 = Yue
year = 2008, to appear
title = Stability of peakons for the Degasperis-Procesi equation
periodical = Comm. Pure Appl. Math.
volume =
issue =
pages =
url = http://aps.arxiv.org/abs/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

*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 = 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
url = http://www.arxiv.org/abs/nlin.SI/0503033
doi = 10.1088/0266-5611/19/6/001

*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
url = http://arxiv.org/abs/nlin.SI/0503036
doi = 10.1155/IMRP.2005.53

*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

*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

*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
url = http://arxiv.org/abs/nlin.SI/0511029
doi = 10.1088/0266-5611/21/6/018

*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
url = http://arxiv.org/abs/nlin.SI/0203055v1
doi = 10.1088/0305-4470/35/22/309

*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

*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

*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