Degasperis–Procesi equation

In mathematical physics, the Degasperis–Procesi equation

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

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

${\displaystyle \displaystyle u_{t}-u_{xxt}+2\kappa u_{x}+(b+1)uu_{x}=bu_{x}u_{xx}+uu_{xxx},}$

where ${\displaystyle \kappa }$ 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 ${\displaystyle \kappa >0}$) 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 ${\displaystyle \kappa =0}$) are the so-called multipeakon solutions, which are functions of the form

${\displaystyle \displaystyle u(x,t)=\sum _{i=1}^{n}m_{i}(t)e^{-|x-x_{i}(t)|}}$

where the functions ${\displaystyle m_{i}}$ and ${\displaystyle x_{i}}$ satisfy^[3]

${\displaystyle {\dot {x}}_{i}=\sum _{j=1}^{n}m_{j}e^{-|x_{i}-x_{j}|},\qquad {\dot {m}}_{i}=2m_{i}\sum _{j=1}^{n}m_{j}\,\operatorname {sgn} {(x_{i}-x_{j})}e^{-|x_{i}-x_{j}|}.}$

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

When ${\displaystyle \kappa >0}$ the soliton solutions of the Degasperis–Procesi equation are smooth; they converge to peakons in the limit as ${\displaystyle \kappa }$ tends to zero.^[5]

Discontinuous solutions

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

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

where ${\displaystyle G(x)=\exp(-|x|)}$, 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 ${\displaystyle u^{2}}$ and ${\displaystyle u_{x}^{2}}$, which only makes sense if u lies in the Sobolev space ${\displaystyle H^{1}=W^{1,2}}$ 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

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; Lundmark & Szmigielski 2005.
5. Matsuno 2005a; Matsuno 2005b.
6. Coclite & Karlsen 2006; Coclite & Karlsen 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., vol. 233, no. 1, pp. 60–91, doi:10.1016/j.jfa.2005.07.008, hdl:10852/10570, S2CID 13339336
• Coclite, Giuseppe Maria; Karlsen, Kenneth Hvistendahl (2007), "On the uniqueness of discontinuous solutions to the Degasperis–Procesi equation" (PDF), J. Differential Equations, vol. 234, no. 1, pp. 142–160, Bibcode:2007JDE...234..142C, doi:10.1016/j.jde.2006.11.008
• 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, arXiv:0709.0905, Bibcode:2009ArRMA.192..165C, doi:10.1007/s00205-008-0128-2, S2CID 17294466
• Degasperis, Antonio; Holm, Darryl D.; Hone, Andrew N. W. (2002), "A new integrable equation with peakon solutions", Theoret. And Math. Phys., vol. 133, no. 2, pp. 1463–1474, arXiv:nlin.SI/0205023, doi:10.1023/A:1021186408422, S2CID 121862973
• Degasperis, Antonio; Procesi, Michela (1999), "Asymptotic integrability", in Degasperis, Antonio; Gaeta, Giuseppe (eds.), Symmetry and Perturbation Theory (Rome, 1998), River Edge, NJ: World Scientific, pp. 23–37
• Dullin, Holger R.; Gottwald, Georg A.; Holm, Darryl D. (2004), "On asymptotically equivalent shallow water wave equations", Physica D, vol. 190, no. 1–2, pp. 1–14, arXiv:nlin.PS/0307011, Bibcode:2004PhyD..190....1D, doi:10.1016/j.physd.2003.11.004, S2CID 16100694
• Escher, Joachim; Liu, Yue; Yin, Zhaoyang (2007), "Shock waves and blow-up phenomena for the periodic Degasperis–Procesi equation", Indiana Univ. Math. J., vol. 56, no. 1, pp. 87–117, doi:10.1512/iumj.2007.56.3040
• Hone, Andrew N. W.; Wang, Jing Ping (2003), "Prolongation algebras and Hamiltonian operators for peakon equations", Inverse Problems, vol. 19, no. 1, pp. 129–145, Bibcode:2003InvPr..19..129H, doi:10.1088/0266-5611/19/1/307, S2CID 250876439
• Ivanov, Rossen (2005), "On the integrability of a class of nonlinear dispersive wave equations", J. Nonlin. Math. Phys., vol. 12, no. 4, pp. 462–468, arXiv:nlin/0606046, Bibcode:2005JNMP...12..462R, doi:10.2991/jnmp.2005.12.4.2, S2CID 248410128
• Ivanov, Rossen (2007), "Water waves and integrability", Phil. Trans. R. Soc. A, vol. 365, no. 1858, pp. 2267–2280, arXiv:0707.1839, Bibcode:2007RSPTA.365.2267I, doi:10.1098/rsta.2007.2007, PMID 17360266, S2CID 11248237
• Johnson, Robin S. (2003), "The classical problem of water waves: a reservoir of integrable and nearly-integrable equations", J. Nonlin. Math. Phys., vol. 10, no. Supplement 1, pp. 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., vol. 17, no. 3, pp. 169–198, Bibcode:2007JNS....17..169L, doi:10.1007/s00332-006-0803-3, S2CID 28451735
• Lundmark, Hans; Szmigielski, Jacek (2003), "Multi-peakon solutions of the Degasperis–Procesi equation", Inverse Problems, vol. 19, no. 6, pp. 1241–1245, arXiv:nlin.SI/0503033, Bibcode:2003InvPr..19.1241L, doi:10.1088/0266-5611/19/6/001, S2CID 250887009
• Lundmark, Hans; Szmigielski, Jacek (2005), "Degasperis–Procesi peakons and the discrete cubic string", Internat. Math. Res. Papers, vol. 2005, no. 2, pp. 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, vol. 21, no. 5, pp. 1553–1570, arXiv:nlin/0511029, Bibcode:2005InvPr..21.1553M, doi:10.1088/0266-5611/21/5/004, S2CID 122820100
• Matsuno, Yoshimasa (2005b), "The N-soliton solution of the Degasperis–Procesi equation", Inverse Problems, vol. 21, no. 6, pp. 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., vol. 35, no. 22, pp. 4775–4790, arXiv:nlin.SI/0203055v1, Bibcode:2002JPhA...35.4775M, doi:10.1088/0305-4470/35/22/309, S2CID 6976529
• Liao, S.J. (2013), "Do peaked solitary water waves indeed exist?", Communications in Nonlinear Science and Numerical Simulation, 19 (6): 1792–1821, arXiv:1204.3354, Bibcode:2014CNSNS..19.1792L, doi:10.1016/j.cnsns.2013.09.042, S2CID 119203215

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., vol. 28, no. 1, pp. 80–105, CiteSeerX 10.1.1.230.4799, doi:10.1093/imanum/drm003
• Escher, Joachim (2007), "Wave breaking and shock waves for a periodic shallow water equation", Phil. Trans. R. Soc. A, vol. 365, no. 1858, pp. 2281–2289, Bibcode:2007RSPTA.365.2281E, doi:10.1098/rsta.2007.2008, PMID 17360268, S2CID 16824268
• Escher, Joachim; Liu, Yue; Yin, Zhaoyang (2006), "Global weak solutions and blow-up structure for the Degasperis–Procesi equation", J. Funct. Anal., vol. 241, no. 2, pp. 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, vol. 368, no. 1–2, pp. 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., vol. 14, no. 3, pp. 390–421, Bibcode:2007JNMP...14..390G, doi:10.2991/jnmp.2007.14.3.8, S2CID 55474222
• Henry, David (2005), "Infinite propagation speed for the Degasperis–Procesi equation", J. Math. Anal. Appl., vol. 311, no. 2, pp. 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" (PDF), Electron. J. Differential Equations, vol. 2007, no. 100, pp. 1–22
• Lenells, Jonatan (2005), "Traveling wave solutions of the Degasperis–Procesi equation", J. Math. Anal. Appl., vol. 306, no. 1, pp. 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., vol. 62, no. 1, pp. 125–146, arXiv:0712.2007, doi:10.1002/cpa.20239, S2CID 7906607
• Liu, Yue; Yin, Zhaoyang (2006), "Global existence and blow-up phenomena for the Degasperis–Procesi equation", Comm. Math. Phys., vol. 267, no. 3, pp. 801–820, Bibcode:2006CMaPh.267..801L, doi:10.1007/s00220-006-0082-5, S2CID 121322682, archived from the original on 2006-10-11
• Liu, Yue; Yin, Zhaoyang (2007), "On the blow-up phenomena for the Degasperis–Procesi equation", Internat. Math. Res. Notices, vol. 2007, doi:10.1093/imrn/rnm117
• Mustafa, Octavian G. (2005), "A note on the Degasperis–Procesi equation", J. Nonlin. Math. Phys., vol. 12, no. 1, pp. 10–14, Bibcode:2005JNMP...12...10M, CiteSeerX 10.1.1.532.782, doi:10.2991/jnmp.2005.12.1.2, S2CID 33488985
• Vakhnenko, Vyacheslav O.; Parkes, E. John (2004), "Periodic and solitary-wave solutions of the Degasperis–Procesi equation" (PDF), Chaos, Solitons and Fractals, vol. 20, no. 5, pp. 1059–1073, Bibcode:2004CSF....20.1059V, doi:10.1016/j.chaos.2003.09.043, archived from the original (PDF) on 2006-09-25
• Yin, Zhaoyang (2003a), "Global existence for a new periodic integrable equation", J. Math. Anal. Appl., vol. 283, no. 1, pp. 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., vol. 47, no. 3, pp. 649–666, archived from the original on December 12, 2012
• Yin, Zhaoyang (2004a), "Global solutions to a new integrable equation with peakons", Indiana Univ. Math. J., vol. 53, no. 4, pp. 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., vol. 212, no. 1, pp. 182–194, doi:10.1016/j.jfa.2003.07.010