Degasperis–Procesi equation

Short description: Used in hydrology

In mathematical physics, the Degasperis–Procesi equation

$u_t-u_{xxt}+2\kappa u_x+4u{u}_x=3u_x{u}_{xx}+u{u}_{xxx}$

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

$u_t-u_{xxt}+2\kappa u_x+(b+1)u{u}_x=b{u}_x{u}_{xx}+u{u}_{xxx},$

where $\kappa$ and b are real parameters (b=3 for the Degasperis–Procesi equation). It was discovered by 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}}}, {{{2}}}) 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.({{{1}}}, {{{2}}})

Soliton solutions

Main article: Peakon

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

$u(x,t)=\sum _{i=1}^n{m}_i(t)e^{-|x-x_i(t)|}$

where the functions $m_i$ and $x_i$ satisfy

${\dot{x}}_i=\sum _{j=1}^n{m}_j{e}^{-|x_i-x_j|},{\dot{m}}_i=2m_i\sum _{j=1}^n{m}_j\mathrm{sgn}(x_i-x_j)e^{-|x_i-x_j|}.$

These ODEs can be solved explicitly in terms of elementary functions, using inverse spectral methods.({{{1}}}, {{{2}}})

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.({{{1}}}, {{{2}}})

Discontinuous solutions

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

${\partial }_{t}u+{\partial }_x[\frac{u^2}{2}+\frac{G}{2}*\frac{3u^2}{2}]=0,$

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).({{{1}}}, {{{2}}}) 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.

References

• Coclite, Giuseppe Maria; Karlsen, Kenneth Hvistendahl (2006), "On the well-posedness of the Degasperis–Procesi equation", J. Funct. Anal. 233 (1): 60–91, doi:10.1016/j.jfa.2005.07.008 
• Coclite, Giuseppe Maria; Karlsen, Kenneth Hvistendahl (2007), "On the uniqueness of discontinuous solutions to the Degasperis–Procesi equation", J. Differential Equations 234 (1): 142–160, doi:10.1016/j.jde.2006.11.008, Bibcode: 2007JDE...234..142C, https://www.math.ntnu.no/conservation/2006/004.pdf 
• Constantin, Adrian; Lannes, David (2007), "The hydrodynamical relevance of the Camassa–Holm and Degasperis–Procesi equations", Archive for Rational Mechanics and Analysis 192 (1): 165–186, doi:10.1007/s00205-008-0128-2, Bibcode: 2009ArRMA.192..165C 
• Degasperis, Antonio; Holm, Darryl D.; Hone, Andrew N. W. (2002), "A new integrable equation with peakon solutions", Theoret. And Math. Phys. 133 (2): 1463–1474, doi:10.1023/A:1021186408422, Bibcode: 2002TMP...133.1463D 
• Degasperis, Antonio; Procesi, Michela (1999), "Asymptotic integrability", in Degasperis, Antonio; Gaeta, Giuseppe, Symmetry and Perturbation Theory (Rome, 1998), River Edge, NJ: World Scientific, pp. 23–37, http://web.tiscalinet.it/SPT2001/SPT98papers/degproc98.ps 
• Dullin, Holger R.; Gottwald, Georg A.; Holm, Darryl D. (2004), "On asymptotically equivalent shallow water wave equations", Physica D 190 (1–2): 1–14, doi:10.1016/j.physd.2003.11.004, Bibcode: 2004PhyD..190....1D 
• Escher, Joachim; Liu, Yue; Yin, Zhaoyang (2007), "Shock waves and blow-up phenomena for the periodic Degasperis–Procesi equation", Indiana Univ. Math. J. 56 (1): 87–117, doi:10.1512/iumj.2007.56.3040, http://www.iumj.indiana.edu/IUMJ/ftdload.php?year=2007&volume=56&artid=3040&ext=pdf 
• Hone, Andrew N. W.; Wang, Jing Ping (2003), "Prolongation algebras and Hamiltonian operators for peakon equations", Inverse Problems 19 (1): 129–145, doi:10.1088/0266-5611/19/1/307, Bibcode: 2003InvPr..19..129H 
• Ivanov, Rossen (2005), "On the integrability of a class of nonlinear dispersive wave equations", J. Nonlin. Math. Phys. 12 (4): 462–468, doi:10.2991/jnmp.2005.12.4.2, Bibcode: 2005JNMP...12..462R 
• Ivanov, Rossen (2007), "Water waves and integrability", Phil. Trans. R. Soc. A 365 (1858): 2267–2280, doi:10.1098/rsta.2007.2007, PMID 17360266, Bibcode: 2007RSPTA.365.2267I 
• Johnson, Robin S. (2003), "The classical problem of water waves: a reservoir of integrable and nearly-integrable equations", J. Nonlin. Math. Phys. 10 (Supplement 1): 72–92, doi:10.2991/jnmp.2003.10.s1.6, Bibcode: 2003JNMP...10S..72J 
• Lundmark, Hans (2007), "Formation and dynamics of shock waves in the Degasperis–Procesi equation", J. Nonlinear Sci. 17 (3): 169–198, doi:10.1007/s00332-006-0803-3, Bibcode: 2007JNS....17..169L, https://dissem.in/p/28857173/formation-and-dynamics-of-shock-waves-in-the-degasperis-procesi-equation 
• Lundmark, Hans; Szmigielski, Jacek (2003), "Multi-peakon solutions of the Degasperis–Procesi equation", Inverse Problems 19 (6): 1241–1245, doi:10.1088/0266-5611/19/6/001, Bibcode: 2003InvPr..19.1241L 
• Lundmark, Hans; Szmigielski, Jacek (2005), "Degasperis–Procesi peakons and the discrete cubic string", International Mathematics Research Papers 2005 (2): 53–116, doi:10.1155/IMRP.2005.53 
• Matsuno, Yoshimasa (2005a), "Multisoliton solutions of the Degasperis–Procesi equation and their peakon limit", Inverse Problems 21 (5): 1553–1570, doi:10.1088/0266-5611/21/5/004, Bibcode: 2005InvPr..21.1553M 
• Matsuno, Yoshimasa (2005b), "The N-soliton solution of the Degasperis–Procesi equation", Inverse Problems 21 (6): 2085–2101, doi:10.1088/0266-5611/21/6/018, Bibcode: 2005InvPr..21.2085M 
• Mikhailov, Alexander V.; Novikov, Vladimir S. (2002), "Perturbative symmetry approach", J. Phys. A: Math. Gen. 35 (22): 4775–4790, doi:10.1088/0305-4470/35/22/309, Bibcode: 2002JPhA...35.4775M 
• Liao (2013), "Do peaked solitary water waves indeed exist?", Communications in Nonlinear Science and Numerical Simulation 19 (6): 1792–1821, doi:10.1016/j.cnsns.2013.09.042, Bibcode: 2014CNSNS..19.1792L 

Further reading

• Coclite, Giuseppe Maria; Karlsen, Kenneth Hvistendahl; Risebro, Nils Henrik (2008), "Numerical schemes for computing discontinuous solutions of the Degasperis–Procesi equation", IMA J. Numer. Anal. 28 (1): 80–105, doi:10.1093/imanum/drm003 
• Escher, Joachim (2007), "Wave breaking and shock waves for a periodic shallow water equation", Phil. Trans. R. Soc. A 365 (1858): 2281–2289, doi:10.1098/rsta.2007.2008, PMID 17360268, Bibcode: 2007RSPTA.365.2281E 
• Escher, Joachim; Liu, Yue; Yin, Zhaoyang (2006), "Global weak solutions and blow-up structure for the Degasperis–Procesi equation", J. Funct. Anal. 241 (2): 457–485, doi:10.1016/j.jfa.2006.03.022 
• Escher, Joachim; Yin, Zhaoyang (2007), "On the initial boundary value problems for the Degasperis–Procesi equation", Phys. Lett. A 368 (1–2): 69–76, doi:10.1016/j.physleta.2007.03.073, Bibcode: 2007PhLA..368...69E 
• Guha, Parta (2007), "Euler–Poincaré formalism of (two component) Degasperis–Procesi and Holm–Staley type systems", J. Nonlin. Math. Phys. 14 (3): 390–421, doi:10.2991/jnmp.2007.14.3.8, Bibcode: 2007JNMP...14..390G 
• Henry, David (2005), "Infinite propagation speed for the Degasperis–Procesi equation", J. Math. Anal. Appl. 311 (2): 755–759, doi:10.1016/j.jmaa.2005.03.001, Bibcode: 2005JMAA..311..755H 
• Hoel, Håkon A. (2007), "A numerical scheme using multi-shockpeakons to compute solutions of the Degasperis–Procesi equation", Electron. J. Differential Equations 2007 (100): 1–22, http://ejde.math.txstate.edu/Volumes/2007/100/hoel.pdf 
• Lenells, Jonatan (2005), "Traveling wave solutions of the Degasperis–Procesi equation", J. Math. Anal. Appl. 306 (1): 72–82, doi:10.1016/j.jmaa.2004.11.038, Bibcode: 2005JMAA..306...72L 
• Lin, Zhiwu; Liu, Yue (2008), "Stability of peakons for the Degasperis–Procesi equation", Comm. Pure Appl. Math. 62 (1): 125–146, doi:10.1002/cpa.20239 
• Liu, Yue; Yin, Zhaoyang (2006), "Global existence and blow-up phenomena for the Degasperis–Procesi equation", Comm. Math. Phys. 267 (3): 801–820, doi:10.1007/s00220-006-0082-5, Bibcode: 2006CMaPh.267..801L, archived from the original on 2006-10-11, https://web.archive.org/web/20061011122559/http://www.mittag-leffler.se/preprints/0506f/info.php?id=22 
• Liu, Yue; Yin, Zhaoyang (2007), "On the blow-up phenomena for the Degasperis–Procesi equation", Internat. Math. Res. Notices 2007, doi:10.1093/imrn/rnm117 
• Mustafa, Octavian G. (2005), "A note on the Degasperis–Procesi equation", J. Nonlin. Math. Phys. 12 (1): 10–14, doi:10.2991/jnmp.2005.12.1.2, Bibcode: 2005JNMP...12...10M 
• Vakhnenko, Vyacheslav O.; Parkes, E. John (2004), "Periodic and solitary-wave solutions of the Degasperis–Procesi equation", Chaos, Solitons and Fractals 20 (5): 1059–1073, doi:10.1016/j.chaos.2003.09.043, Bibcode: 2004CSF....20.1059V, http://www.maths.strath.ac.uk/~caas35/v&pCSF04.pdf 
• Yin, Zhaoyang (2003a), "Global existence for a new periodic integrable equation", J. Math. Anal. Appl. 283 (1): 129–139, doi:10.1016/S0022-247X(03)00250-6 
• Yin, Zhaoyang (2003b), "On the Cauchy problem for an integrable equation with peakon solutions", Illinois J. Math. 47 (3): 649–666, http://www.math.uiuc.edu/~hildebr/ijm/fall03/final/yin.html 
• Yin, Zhaoyang (2004a), "Global solutions to a new integrable equation with peakons", Indiana Univ. Math. J. 53 (4): 1189–1209, doi:10.1512/iumj.2004.53.2479 
• Yin, Zhaoyang (2004b), "Global weak solutions for a new periodic integrable equation with peakon solutions", J. Funct. Anal. 212 (1): 182–194, doi:10.1016/j.jfa.2003.07.010