  We study the periodic Cauchy problem for an integrable equation with cubic
nonlinearities introduced by V. Novikov. Like the Camassa-Holm and
Degasperis-Procesi equations, Novikov's equation has Lax pair representations
and admits peakon solutions, but it has nonlinear terms that are cubic, rather
than quadratic. We show the local well-posedness of the problem in Sobolev
spaces and existence and uniqueness of solutions for all time using orbit
invariants. Furthermore we prove a Cauchy-Kowalevski type theorem for this
equation, that establishes the existence and uniqueness of real analytic
solutions.

Analysis of PDEs