Recent generalizations of the Camassa-Holm equation are studied from the point of view of existence of global solutions, criteria for wave breaking phenomena and integrability. We provide conditions, based on lower bounds for the first spatial derivative of local solutions, for global well-posedness for the family under consideration in Sobolev spaces. Moreover, we prove that wave breaking phenomena occurs under certain mild hypothesis. Regarding integrability, we apply the machinery developed by Dubrovin [Commun. Math. Phys. 267, 117--139 (2006)] to prove that there exists a unique bi-hamiltonian structure for the equation only when it is reduced to the Dullin-Gotwald-Holm equation. Our results suggest that a recent shallow water model incorporating Coriollis efects is integrable only in specific situations. Finally, to finish the scheme of geometric integrability of the family of equations initiated in a previous work, we prove that the Dullin-Gotwald-Holm equation describes pseudo-spherical surfaces.

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Camassa-Holm 方程推广的可积性、全局解的存在性和波浪破坏标准

从全局解的存在性、波浪破碎现象的标准和可积性的角度研究了 Camassa-Holm 方程的最新推广。我们根据局部解的一阶空间导数的下界，为 Sobolev 空间中考虑的族的全局适定性提供条件。此外，我们证明了在某些温和的假设下会发生破浪现象。关于可集成性，我们应用了 Dubrovin [Commun. 数学。物理。267, 117--139 (2006)] 证明该方程只有在简化为 Dullin-Gotwald-Holm 方程时才存在独特的双哈密尔顿结构。我们的结果表明，最近的包含科里奥利效应的浅水模型仅在特定情况下是可集成的。最后，