SIAM Journal on Mathematical Analysis, Volume 53, Issue 3, Page 3016-3039, January 2021.
Peaked periodic waves in the Camassa--Holm equation are revisited. Linearized evolution equations are derived for perturbations to the peaked periodic waves, and linearized instability is proven in both $H^1$ and $W^{1,\infty}$ norms. Dynamics of perturbations in $H^1$ is related to the existence of two conserved quantities and is bounded in the full nonlinear system due to these conserved quantities. On the other hand, perturbations to the peaked periodic wave grow in the $W^{1,\infty}$ norm and may blow up in a finite time in the nonlinear evolution of the Camassa--Holm equation.

中文翻译：

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Camassa-Holm方程中对峰值周期波的摄动增长

SIAM数学分析杂志，第53卷，第3期，第3016-3039页，2021年1月
。再次讨论了Camassa-Holm方程中的峰值周期波。推导了线性化演化方程，以扰动峰值周期波，并在$ H ^ 1 $和$ W ^ {1，\ infty} $范数中证明了线性化的不稳定性。$ H ^ 1 $中的摄动动力学与两个守恒量的存在有关，并且由于这些守恒量而在完整的非线性系统中有界。另一方面，对峰值周期波的扰动在$ W ^ {1，\ infty} $范数中增长，并且可能在有限的时间内在Camassa-Holm方程的非线性演化中爆炸。