Abstract
Peakon traveling wave solutions on the circle are derived for the cubic ab-family of equations, which includes both the Fokas–Olver–Rosenau–Qiao (FORQ) and Novikov (NE) equations. For a≠0, it is proved that there exists an initial data in the Sobolev space Hs, s<32, with nonunique solutions on the circle. We construct a two-peakon solution with an asymmetric peakon–antipeakon initial profile that collides at a finite time. At the time of collision, the two-peakon solution reduces to a single peakon solution, which will complete the proof of nonuniqueness.
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