In this paper, we study the orbital stability of peakons and periodic peakons for a nonlinear quartic Camassa-Holm equation. We first verify that the QCHE has global peakon and periodic peakon solutions. Then by the invariants of the equation and controlling the extrema of the solution, we prove that the shapes of the peakons and periodic peakons are stable under small perturbations in the energy space.

中文翻译：

---

非线性四次 Camassa-Holm 方程峰子和周期峰子的稳定性

在本文中，我们研究了非线性四次 Camassa-Holm 方程的峰子和周期峰子的轨道稳定性。我们首先验证 QCHE 具有全局峰值和周期性峰值解决方案。然后通过方程的不变量和控制解的极值，我们证明了在能量空间的小扰动下峰子和周期峰子的形状是稳定的。