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

José Manuel Palacios, ,Institut Denis Poisson, Université de Tours, Université d'Orleans, CNRS, Parc Grandmont 37200, Tours, France
2019 Discrete and Continuous Dynamical Systems. Series A  
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
more » ... 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. 2475 2476 JOSÉ MANUEL PALACIOS Moreover, by using this matrix Lax-pair representation, Hone and Wang showed how the Novikov equation is related by a reciprocal transformation to a negative flow in the Sawada-Kotera hierarchy. The Novikov equation possesses infinitely many conservation laws, among which, the most important ones are given by E(u) :=ˆR u 2 (t, x) + u 2 x (t, x) dx and F (u) :=ˆ u 4 + 2u 2 u 2 x −
doi:10.3934/dcds.2020372 fatcat:ak4rtbmztbfxto66irphlwb5fe