Nonuniqueness for the \(ab\)-family of equations with peakon travelling waves on the circle. (English) Zbl 1492.35231

Puri, Rajan

Rocky Mt. J. Math. 52, No. 2, 707-715 (2022).

Summary: 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 \(a \neq 0\), it is proved that there exists an initial data in the Sobolev space \(H^s, s < \frac{3}{2}\), 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.

MSC:

35Q35

PDEs in connection with fluid mechanics

Keywords:

well-posedness; initial value problem; Cauchy problem; Sobolev spaces; Camassa-Holm equation; solitons; peakon

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