$W^{1,\infty}$ instability of $H^1$-stable peakons in the Novikov equation

TL;DR

This paper demonstrates that while Novikov equation peakons are stable in $H^1$, they become unstable under $W^{1,inity}$-perturbations, which can cause solutions to blow up in finite time.

Contribution

It reveals the $W^{1,inity}$ instability of $H^1$-stable peakons in the Novikov equation using the method of characteristics.

Findings

Peakons are unstable in $W^{1,inity}$

Small $W^{1,inity}$ perturbations can cause finite time blow-up

Stability in $H^1$ does not imply stability in $W^{1,inity}$

Abstract

It is known from the previous works that the peakon solutions of the Novikov equation are orbitally and asymptotically stable in $H^1$. We prove, via the method of characteristics, that these peakon solutions are unstable under $W^{1,\infty}$-perturbations. Moreover, we show that small initial $W^{1,\infty}$-perturbations of the Novikov peakons can lead to the finite time blow-up of the corresponding solutions.