This paper is devoted to the new shallow-water model (also called the modified Camassa–Holm–Novikov equation) with cubic nonlinearity, which admits the single peaked solitons and multi-peakon solutions, and includes both the Fokas–Olver–Rosenau–Qiao equation and the Novikov equation as two special cases. It is shown that the Cauchy problem of the modified Camassa–Holm–Novikov equation for the periodic and the nonperiodic case is well-posed in Sobolev spaces in the sense of Hadamard, that is, the data-to-solution map is continuous. However, the solution map is not uniformly continuous.

中文翻译：

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具有立方非线性的新浅水模型的适定性和连续性

本文致力于具有立方非线性的新浅水模型（也称为修正的Camassa–Holm–Novikov方程），该模型允许单峰孤子和多峰解​​，并且包括Fokas–Olver–Rosenau–Qiao方程和Novikov方程是两种特殊情况。结果表明，修正的Camassa–Holm–Novikov方程针对周期和非周期情况的柯西问题在Hadamard的意义上在Sobolev空间中具有很好的位置，也就是说，数据到解的映射是连续的。但是，解图不是一致连续的。