Degasperis–Procesi equation

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In mathematical physics, the Degasperis–Procesi equation

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

where 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 ) 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 ) are the so-called multipeakon solutions, which are functions of the form

where the functions and satisfy[3]

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

When the soliton solutions of the Degasperis–Procesi equation are smooth; they converge to peakons in the limit as tends to zero.[5]

Discontinuous solutions

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

where , 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 and , which only makes sense if u lies in the Sobolev space 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 [permanent dead link]
  • 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 [permanent dead link]
  • Constantin, Adrian; Lannes, David (2007), The hydrodynamical relevance of the Camassa–Holm and Degasperis–Procesi equations, Bibcode:2009ArRMA.192..165C, arXiv:0709.0905Freely accessible, 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:nlin.SI/0205023Freely accessible, 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, Bibcode:2004PhyD..190....1D, arXiv:nlin.PS/0307011Freely accessible, 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, Bibcode:2007RSPTA.365.2267I, arXiv:0707.1839Freely accessible, 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, Bibcode:2003InvPr..19.1241L, arXiv:nlin.SI/0503033Freely accessible, 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:nlin.SI/0503036Freely accessible, 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, Bibcode:2005InvPr..21.1553M, arXiv:nlin/0511029Freely accessible, 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, Bibcode:2005InvPr..21.2085M, arXiv:nlin.SI/0511029Freely accessible, 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, Bibcode:2002JPhA...35.4775M, arXiv:nlin.SI/0203055v1Freely accessible, 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, Bibcode:2014CNSNS..19.1792L, arXiv:1204.3354Freely accessible, doi:10.1016/j.cnsns.2013.09.042 

Further reading

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