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Zbl 1153.35075
Hone, Andrew N.W.; Wang, Jing Ping
Integrable peakon equations with cubic nonlinearity. (English)
[J] J. Phys. A, Math. Theor. 41, No. 37, Article ID 372002, 10 p. (2008). ISSN 1751-8113; ISSN 1751-8121/e

Summary: We present a new integrable partial differential equation found by Vladimir Novikov. Like the Camassa-Holm and Degasperis-Procesi equations, this new equation admits peaked soliton (peakon) solutions, but it has nonlinear terms that are cubic, rather than quadratic. We give a matrix Lax pair for V. Novikov's equation, and show how it is related by a reciprocal transformation to a negative flow in the Sawada-Kotera hierarchy. Infinitely many conserved quantities are found, as well as a bi-Hamiltonian structure. The latter is used to obtain the Hamiltonian form of the finite-dimensional system for the interaction of $N$ peakons, and the two-body dynamics $(N = 2)$ is explicitly integrated. Finally, all of this is compared with some analogous results for another cubic peakon equation derived by Zhijun Qiao.

MSC 2000:: *35Q58 Other completely integrable PDE
35Q51 Solitons
37K10 Completely integrable systems etc.

Keywords: Novikov's equation; peakon; Lax pair; Sawada-Katera hierarchy; Hamiltonian form

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Zentralblatt MATH Berlin [Germany]