论文标题
$ b $ fimy of Pearpemon方程的稳定性
The Stability of the $b$-family of Peakon Equations
论文作者
论文摘要
在目前的工作中，我们重新访问了PearpeN方程的$ B $家庭模型，其中包含特殊情况，$ B = 2 $（Camassa-Holm）和$ B = 3 $（Degasperis-Procesi）可集成的示例。我们建立了有关Pearpon Solutions的点频谱的信息，尤其是发现，对于合适的平滑扰动，右半平面中存在点频谱，使Peardons呈现不稳定的$ B <1 $。我们在定点迭代的领域，光谱稳定性分析和不同参数制度模型的时间步长的领域进行数值探索。特别是，我们以$ b <-1 $的价格确定了精确的，固定的（光谱稳定）的Lefton解决方案，并且对于$ -1 <b <1 $，我们将Ramp-Cliff Solutions识别为该制度中的主要国家。我们通过检查平滑初始数据将平滑的初始数据分解为稳定的Peakons，以$ b> 1 $进行补充。尽管在较早的研究中探索了上述许多动力学特征，但在当前的工作中，我们会尽可能通过光谱稳定性计算来补充它们。
In the present work we revisit the $b$-family model of peakon equations, containing as special cases the $b=2$ (Camassa-Holm) and $b=3$ (Degasperis-Procesi) integrable examples. We establish information about the point spectrum of the peakon solutions and notably find that for suitably smooth perturbations there exists point spectrum in the right half plane rendering the peakons unstable for $b<1$. We explore numerically these ideas in the realm of fixed-point iterations, spectral stability analysis and time-stepping of the model for the different parameter regimes. In particular, we identify exact, stationary (spectrally stable) lefton solutions for $b<-1$, and for $-1<b<1$, we dynamically identify ramp-cliff solutions as dominant states in this regime. We complement our analysis by examining the breakup of smooth initial data into stable peakons for $b>1$. While many of the above dynamical features had been explored in earlier studies, in the present work, we supplement them, wherever possible, with spectral stability computations.
