Geodesic
A~New Integrable Equation with Peakon Solutions
Teoretičeskaâ i matematičeskaâ fizika, Tome 133 (2002) no. 2, pp. 170-183

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We consider a new partial differential equation recently obtained by Degasperis and Procesi using the method of asymptotic integrability; this equation has a form similar to the Camassa–Holm shallow water wave equation. We prove the exact integrability of the new equation by constructing its Lax pair and explain its relation to a negative flow in the Kaup–Kupershmidt hierarchy via a reciprocal transformation. The infinite sequence of conserved quantities is derived together with a proposed bi-Hamiltonian structure. The equation admits exact solutions as a superposition of multipeakons, and we describe the integrable finite-dimensional peakon dynamics and compare it with the analogous results for Camassa–Holm peakons.
Keywords: peakons, reciprocal transformations, weak solutions. 
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     author = {A. Degasperis and D. D. Holm and A. Hone},
     title = {A~New {Integrable} {Equation} with {Peakon} {Solutions}},
     journal = {Teoreti\v{c}eska\^a i matemati\v{c}eska\^a fizika},
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     url = {http://geodesic.mathdoc.fr/item/TMF_2002_133_2_a3/}
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A. Degasperis; D. D. Holm; A. Hone. A~New Integrable Equation with Peakon Solutions. Teoretičeskaâ i matematičeskaâ fizika, Tome 133 (2002) no. 2, pp. 170-183. http://geodesic.mathdoc.fr/item/TMF_2002_133_2_a3/