Non-uniqueness for the ab-family of equations

Abstract

We study the cubic ab-family of equations, which includes both the Fokas-Olver-Rosenau-Qiao (FORQ) and the Novikov (NE) equations. For $a\ne 0$, it is proved that there exist initial data in the Sobolev space $H^s$, $s<3/2$, with non-unique solutions. Multiple solutions are constructed by studying the collision of 2-peakon solutions. Furthermore, we prove the novel phenomenon that for some members of the family, collision between 2-peakons can occur even if the "faster" peakon is in front of the "slower" peakon.