  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.

Exactly Solvable and Integrable Systems;Pattern Formation and Solitons