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

Discontinuous solutions

The Degasperis–Procesi equation (with $\kappa=0$ ) 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 $u^2$ and $u_x^2$ , which only makes sense if u lies in the Sobolev space $H^1 = W^{1,2}$ 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

↑ 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, 2005
↑ Matsuno 2005a, 2005b
↑ 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: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), pp. 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
Dullin, Holger R.; Gottwald, Georg A.; Holm, Darryl D. (2004), "On asymptotically equivalent shallow water wave equations", Physica D 190, pp. 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), 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:0707.1839, 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: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), 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 21 (5), pp. 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), 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. 35 (22), pp. 4775–4790, arXiv:nlin.SI/0203055v1, 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:1204.3354, Bibcode:2014CNSNS..19.1792L, doi:10.1016/j.cnsns.2013.09.042

Degasperis–Procesi equation

Soliton solutions

Discontinuous solutions

Notes

References

Further reading