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Integrable peakon equations with cubic nonlinearity. (English) Zbl 1153.35075
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 $\left(N=2\right)$ is explicitly integrated. Finally, all of this is compared with some analogous results for another cubic peakon equation derived by Zhijun Qiao.
MSC:
 35Q58 Other completely integrable PDE (MSC2000) 35Q51 Soliton-like equations 37K10 Completely integrable systems, integrability tests, bi-Hamiltonian structures, hierarchies