Degasperis–Procesi equation

In mathematical physics, the Degasperis–Procesi equation

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

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

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

where κ 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.^[1] 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.^[2]

Soliton solutions

Main article: Peakon

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

$\displaystyle u(x,t)=\sum_{i=1}^n m_i(t) e^{-|x-x_i(t)|}$

where the functions m_i and x_i satisfy^[3]

$\dot{x}_i = \sum_{j=1}^n m_j e^{-|x_i-x_j|},\qquad \dot{m}_i = 2 m_i \sum_{j=1}^n m_j\, \sgn{(x_i-x_j)} e^{-|x_i-x_j|}.$

These ODEs can be solved explicitly in terms of elementary functions, using inverse spectral methods.^[4]

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.^[5]

Discontinuous solutions

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

$\partial_t u + \partial_x \left[\frac{u^2}{2} + \frac{G}{2} * \frac{3 u^2}{2} \right] = 0,$

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 u² and $u_x^2$, which only makes sense if u lies in the Sobolev space H¹ = 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

1. ^ Degasperis & Procesi 1999; Degasperis, Holm & Hone 2002; Mikhailov & Novikov 2002; Hone & Wang 2003; Ivanov 2005
2. ^ Johnson 2003; Dullin, Gottwald & Holm 2004; Constantin & Lannes 2007; Ivanov 2007
3. ^ Degasperis, Holm & Hone 2002
4. ^ Lundmark & Szmigielski 2003, 2005
5. ^ Matsuno 2005a, 2005b
6. ^ Coclite & Karlsen 2006, 2007; Lundmark 2007; Escher, Liu & Yin 2007

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, http://www.math.uio.no/~kennethk/articles/art113_journal.pdf 
• 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, http://www.math.uio.no/~kennethk/articles/art122_journal.pdf 
• Constantin, Adrian; Lannes, David (2007), 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 
• 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, arXiv:nlin.SI/0205023, doi:10.1023/A:1021186408422 
• 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: 1–14, arXiv:nlin.PS/0307011, Bibcode 2004PhyD..190....1D, doi:10.1016/j.physd.2003.11.004 
• 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, 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, Bibcode 2003InvPr..19..129H, doi:10.1088/0266-5611/19/1/307 
• Ivanov, Rossen (2005), "On the integrability of a class of nonlinear dispersive wave equations", J. Nonlin. Math. Phys. 12 (4): 462–468, Bibcode 2005JNMP...12..462R, doi:10.2991/jnmp.2005.12.4.2 
• Ivanov, Rossen (2007), "Water waves and integrability", Phil. Trans. R. Soc. A 365 (1858): 2267–2280, Bibcode 2007RSPTA.365.2267I, doi:10.1098/rsta.2007.2007 
• 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, Bibcode 2003JNMP...10S..72J, doi:10.2991/jnmp.2003.10.s1.6 
• Lundmark, Hans (2007), "Formation and dynamics of shock waves in the Degasperis–Procesi equation", J. Nonlinear Sci. 17 (3): 169–198, Bibcode 2007JNS....17..169L, doi:10.1007/s00332-006-0803-3, http://www.mittag-leffler.se/preprints/0506f/info.php?id=26 
• Lundmark, Hans; Szmigielski, Jacek (2003), "Multi-peakon solutions of the Degasperis–Procesi equation", Inverse Problems 19 (6): 1241–1245, arXiv:nlin.SI/0503033, Bibcode 2003InvPr..19.1241L, doi:10.1088/0266-5611/19/6/001 
• Lundmark, Hans; Szmigielski, Jacek (2005), "Degasperis–Procesi peakons and the discrete cubic string", Internat. Math. Res. Papers 2005 (2): 53–116, arXiv:nlin.SI/0503036, 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, arXiv:nlin/0511029, Bibcode 2005InvPr..21.1553M, doi:10.1088/0266-5611/21/5/004 
• Matsuno, Yoshimasa (2005b), "The N-soliton solution of the Degasperis–Procesi equation", Inverse Problems 21 (6): 2085–2101, arXiv:nlin.SI/0511029, Bibcode 2005InvPr..21.2085M, doi:10.1088/0266-5611/21/6/018 
• Mikhailov, Alexander V.; Novikov, Vladimir S. (2002), "Perturbative symmetry approach", J. Phys. A: Math. Gen. 35 (22): 4775–4790, arXiv:nlin.SI/0203055v1, Bibcode 2002JPhA...35.4775M, doi:10.1088/0305-4470/35/22/309 

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, http://www.math.uio.no/~kennethk/articles/art125.pdf 
• Escher, Joachim (2007), "Wave breaking and shock waves for a periodic shallow water equation", Phil. Trans. R. Soc. A 365 (1858): 2281–2289, Bibcode 2007RSPTA.365.2281E, doi:10.1098/rsta.2007.2008 
• 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, Bibcode 2007PhLA..368...69E, doi:10.1016/j.physleta.2007.03.073 
• Guha, Parta (2007), "Euler–Poincaré formalism of (two component) Degasperis–Procesi and Holm–Staley type systems", J. Nonlin. Math. Phys. 14 (3): 390–421, Bibcode 2007JNMP...14..390G, doi:10.2991/jnmp.2007.14.3.8 
• Henry, David (2005), "Infinite propagation speed for the Degasperis–Procesi equation", J. Math. Anal. Appl. 311 (2): 755–759, Bibcode 2005JMAA..311..755H, doi:10.1016/j.jmaa.2005.03.001 
• 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, Bibcode 2005JMAA..306...72L, doi:10.1016/j.jmaa.2004.11.038 
• Lin, Zhiwu; Liu, Yue (2008), "Stability of peakons for the Degasperis–Procesi equation", Comm. Pure Appl. Math. 62 (1): 125–146, arXiv:0712.2007, 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, Bibcode 2006CMaPh.267..801L, doi:10.1007/s00220-006-0082-5, 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, Bibcode 2005JNMP...12...10M, doi:10.2991/jnmp.2005.12.1.2 
• 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, Bibcode 2004CSF....20.1059V, doi:10.1016/j.chaos.2003.09.043, 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