Degasperis–Procesi equation
In mathematical physics, the Degasperis–Procesi equation
is one of only two
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
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
Notes
- ^ Degasperis & Procesi 1999; Degasperis, Holm & Hone 2002; Mikhailov & Novikov 2002; Hone & Wang 2003; Ivanov 2005.
- ^ Johnson 2003; Dullin, Gottwald & Holm 2004; Constantin & Lannes 2007; Ivanov 2007.
- ^ Degasperis, Holm & Hone 2002.
- ^ Lundmark & Szmigielski 2003; Lundmark & Szmigielski 2005.
- ^ Matsuno 2005a; Matsuno 2005b.
- ^ 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, 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,
- 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, 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, 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, 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,
- 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, 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, S2CID 248410128
- Ivanov, Rossen (2007), "Water waves and integrability", Phil. Trans. R. Soc. A, vol. 365, no. 1858, pp. 2267–2280, 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,
- Lundmark, Hans (2007), "Formation and dynamics of shock waves in the Degasperis–Procesi equation", J. Nonlinear Sci., vol. 17, no. 3, pp. 169–198, S2CID 28451735
- Lundmark, Hans; Szmigielski, Jacek (2003), "Multi-peakon solutions of the Degasperis–Procesi equation", Inverse Problems, vol. 19, no. 6, pp. 1241–1245, 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,
- Matsuno, Yoshimasa (2005a), "Multisoliton solutions of the Degasperis–Procesi equation and their peakon limit", Inverse Problems, vol. 21, no. 5, pp. 1553–1570, S2CID 122820100
- Matsuno, Yoshimasa (2005b), "The N-soliton solution of the Degasperis–Procesi equation", Inverse Problems, vol. 21, no. 6, pp. 2085–2101,
- Mikhailov, Alexander V.; Novikov, Vladimir S. (2002), "Perturbative symmetry approach", J. Phys. A: Math. Gen., vol. 35, no. 22, pp. 4775–4790, S2CID 6976529
- 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,
- 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, 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,
- 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,
- 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, S2CID 55474222
- Henry, David (2005), "Infinite propagation speed for the Degasperis–Procesi equation", J. Math. Anal. Appl., vol. 311, no. 2, pp. 755–759,
- 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,
- Lin, Zhiwu; Liu, Yue (2008), "Stability of peakons for the Degasperis–Procesi equation", Comm. Pure Appl. Math., vol. 62, no. 1, pp. 125–146, 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, S2CID 121322682, archived from the originalon 2006-10-11
- Liu, Yue; Yin, Zhaoyang (2007), "On the blow-up phenomena for the Degasperis–Procesi equation", Internat. Math. Res. Notices, vol. 2007,
- Mustafa, Octavian G. (2005), "A note on the Degasperis–Procesi equation", J. Nonlin. Math. Phys., vol. 12, no. 1, pp. 10–14, 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, 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,
- 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,
- 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,