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 derived by Zhijun Qiao.Submitted to: J. Phys. A: Math. Gen.

We consider a new partial differential equation, of a similar form to the Camassa-Holm shallow water wave equation, which was recently obtained by Degasperis and Procesi using the method of asymptotic integrability. We prove the exact integrability of the new equation by constructing its Lax pair, and we explain its connection with 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 in the form of a superposition of multi-peakons, and we describe the integrable finite-dimensional peakon dynamics and compare it with the analogous results for Camassa-Holm peakons.

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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.

We consider nonlinear recurrences generated from cluster mutations applied to quivers that have the property of being cluster mutation-periodic with period 1. Such quivers were completely classified by Fordy and Marsh, who characterised them in terms of the skew-symmetric matrix that defines the quiver. The associated nonlinear recurrences are equivalent to birational maps, and we explain how these maps can be endowed with an invariant Poisson bracket and/or presymplectic structure.Upon applying the algebraic entropy test, we are led to a series of conjectures which imply that the entropy of the cluster maps can be determined from their tropical analogues, which leads to a sharp classification result. Only four special families of these maps should have zero entropy. These families are examined in detail, with many explicit examples given, and we show how they lead to discrete dynamics that is integrable in the Liouville-Arnold sense.