We study the cubic ab-family of equations, which includes both the Fokas-Olver-Rosenau-Qiao (FORQ) and the Novikov (NE) equations. For $a\neq0$, 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.

中文翻译：

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ab族方程的非唯一性

我们研究三次 ab 族方程，其中包括 Fokas-Olver-Rosenau-Qiao (FORQ) 和 Novikov (NE) 方程。对于$a\neq0$，证明在Sobolev空间$H^s$中存在初始数据$s<3/2$，且具有非唯一解。通过研究 2-peakon 解的碰撞来构建多个解。此外，我们证明了一种新现象，即对于某些家庭成员，即使“较快”的峰在“较慢”的峰之前，也可能发生 2 峰之间的碰撞。