Nonlinear Physics 2003
DOI: 10.1142/9789812704467_0005
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Integrable and Non-Integrable Equations With Peakons

Abstract: We consider a one-parameter family of non-evolutionary partial differential equations which includes the integrable Camassa-Holm equation and a new integrable equation first isolated by Degasperis and Procesi. A Lagrangian and Hamiltonian formulation is presented for the whole family of equations, and we discuss how this fits into a bi-Hamiltonian framework in the integrable cases. The Hamiltonian dynamics of peakons and some other special finite-dimensional reductions are also described.

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Cited by 109 publications
(171 citation statements)
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“…In fact Kupershmidt [29] [7,8] using the asymptotic integrability method. The second member of this one-parameter family of PDEs is called Degasperis-Procesi equation [6].…”
Section: Motivation Results and Planmentioning
confidence: 99%
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“…In fact Kupershmidt [29] [7,8] using the asymptotic integrability method. The second member of this one-parameter family of PDEs is called Degasperis-Procesi equation [6].…”
Section: Motivation Results and Planmentioning
confidence: 99%
“…One can easily use (8) to compute the second Hamiltonian operator O KdV of the KdV equation which corresponds to the action of Vect(S 1 ) on the extended ψD(S 1 ) given by:…”
Section: Theorem 2 the Euler-poincaré Flow Which Is Induced By The Acmentioning
confidence: 99%
See 2 more Smart Citations
“…Apart from some limiting cases, the general odd solution of (1.15) turns out to be [19]. In the next section we derive the functional equation from the Jacobi identity for the operator (1.8), and then proceed to show how this leads to a suitable bracket for the peakons.…”
Section: Introductionmentioning
confidence: 99%