Orbital and asymptotic stability of a train of peakons for the Novikov equation

TL;DR

This paper proves the orbital and asymptotic stability of peakon and multi-peakon solutions for the Novikov equation, improving stability results and removing previous non-negativity assumptions.

Contribution

It establishes stability of peakon trains for the Novikov equation and extends stability results to cases without non-negativity constraints on initial momentum.

Findings

Proved orbital and asymptotic stability of peakon trains.

Improved stability results for single peakons without non-negativity.

Extended stability to non-ordered multi-peakons.

Abstract

The Novikov equation is an integrable Camassa-Holm type equation with cubic nonlinearity. One of the most important features of this equation is the existence of peakon and multi-peakon solutions, i.e. peaked traveling waves behaving as solitons. This paper aims to prove both, the orbital and asymptotic stability of peakon trains solutions, i.e. multi-peakon solutions such that their initial configuration is increasingly ordered. Furthermore, we give an improvement of the orbital stability of a single peakon so that we can drop the non-negativity hypothesis on the momentum density. The same result also holds for the orbital stability for peakon trains, i.e. in this latter case we can also avoid assuming non-negativity of the initial momentum density. Finally, as a corollary of these results together with some asymptotic formulas for the position and momenta vectors for multi-peakon solutions, we obtain the orbital and asymptotic stability for initially not well-ordered multipeakons.