# 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 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 ${\displaystyle \kappa >0}$) has later been found to play a similar role in water wave theory as the Camassa–Holm equation.[2]

## Soliton solutions

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, 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" (PDF), J. Funct. Anal., 233 (1), pp. 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" (PDF), J. Differential Equations, 234 (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, arXiv:, 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), pp. 1463–1474, arXiv:, 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
• Dullin, Holger R.; Gottwald, Georg A.; Holm, Darryl D. (2004), "On asymptotically equivalent shallow water wave equations", Physica D, 190, pp. 1–14, arXiv:, 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), pp. 87–117
• Hone, Andrew N. W.; Wang, Jing Ping (2003), "Prolongation algebras and Hamiltonian operators for peakon equations", Inverse Problems, 19 (1), pp. 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), pp. 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), pp. 2267–2280, arXiv:, 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), 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., 17 (3), pp. 169–198, Bibcode:2007JNS....17..169L, doi:10.1007/s00332-006-0803-3
• Lundmark, Hans; Szmigielski, Jacek (2003), "Multi-peakon solutions of the Degasperis–Procesi equation", Inverse Problems, 19 (6), pp. 1241–1245, arXiv:, 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), pp. 53–116, arXiv:, doi:10.1155/IMRP.2005.53
• Matsuno, Yoshimasa (2005a), "Multisoliton solutions of the Degasperis–Procesi equation and their peakon limit", Inverse Problems, 21 (5), pp. 1553–1570, arXiv:, 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), pp. 2085–2101, arXiv:, 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), pp. 4775–4790, arXiv:, Bibcode:2002JPhA...35.4775M, doi:10.1088/0305-4470/35/22/309
• Liao, S.J. (2013), "Do peaked solitary water waves indeed exist?", Communications in Nonlinear Science and Numerical Simulation, arXiv:, Bibcode:2014CNSNS..19.1792L, doi:10.1016/j.cnsns.2013.09.042