alexa Integrable peakon equations with cubic nonlinearity
Mathematics

Mathematics

Journal of Generalized Lie Theory and Applications

Author(s): Andrew N W Hone, Jing Ping Wang

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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.

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This article was published in Journal of Physics A: Mathematical and Theoretical and referenced in Journal of Generalized Lie Theory and Applications

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