The periodic Cauchy problem for Novikov's equation

TL;DR

This paper investigates the well-posedness and analytic solutions of Novikov's integrable equation with cubic nonlinearities, extending the understanding of peakon solutions and orbit invariants in nonlinear wave equations.

Contribution

It establishes local and global well-posedness, and proves a Cauchy-Kowalevski theorem for Novikov's equation, highlighting its analytic solutions and invariants.

Findings

Proves local well-posedness in Sobolev spaces

Establishes global existence and uniqueness of solutions

Demonstrates existence of real analytic solutions

Abstract

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.