A nonlinear generalization of the Camassa-Holm equation with peakon solutions

Instability of $H^1$-stable peakons in the Camassa-Holm equation

It is well-known that peakons in the Camassa-Holm equation are $H^1$-orbitally stable thanks to the presence of conserved quantities and properties of peakons as constrained energy minimizers. By using the method of characteristics, we prove that piecewise $C^1$ perturbations to peakons grow in time in spite of their stability in the $H^1$-norm. We also show that the linearized stability analysis near peakons contradicts the $H^1$-orbital stability result, hence passage from linear to nonlinear theory is false in $H^1$.

具有变系数和奇异摄动的修正 Camassa-Holm 方程的渐近类孤子解与渐近类峰值子解

Asymptotic soliton-like and asymptotic peakon-like solutions of the modified Camassa-Holm equation with variable coefficients and singular perturbation

针对带变系数与奇异摄动的修正 Camassa-Holm 方程，提出任意阶渐近类孤子与类峰值子解构造方案，并通过实例验证其精度与可推广性。

An integrable semi-discretization of the Camassa-Holm equation and its determinant solution

An integrable semi-discretization of the Camassa-Holm equation is presented. The keys of its construction are bilinear forms and determinant structure of solutions of the CH equation. Determinant formulas of $N$-soliton solutions of the continuous and semi-discrete Camassa-Holm equations are presented. Based on determinant formulas, we can generate multi-soliton, multi-cuspon and multi-soliton-cuspon solutions. Numerical computations using the integrable semi-discrete Camassa-Holm equation are performed. It is shown that the integrable semi-discrete Camassa-Holm equation gives very accurate nu…

Orbital stability of peakon solutions for a generalized higher-order Camassa-Holm equation

In this paper, we investigate the orbital stability issue of a generalized higher-order Camassa-Holm (HOCH) equation, which is an higher-order extension of the quadratic CH equation. Firstly, we show that the HOCH equation admits a global weak peakon solution by paring it with ssome smooth test function. Secondly, with the help of two conserved quantities and the non-sgn-changing condition, we prove the orbital stability of this peakon solution in the energy space in the sense that its shape remains approximately the same for all times. Our results enrich the research of the orbital stability …

Spectral instability of peakons in the b-family of the Camassa--Holm equations

We prove spectral instability of peakons in the $b$-family of Camassa--Holm equations in $L^2(\mathbb{R})$ that includes the integrable cases of $b = 2$ and $b = 3$. We start with a linearized operator defined on functions in $H^1(\mathbb{R}) \cap W^{1,\infty}(\mathbb{R})$ and extend it to a linearized operator defined on weaker functions in $L^2(\mathbb{R})$. For $b \neq \frac{5}{2}$, the spectrum of the linearized operator in $L^2(\mathbb{R})$ is proved to cover a closed vertical strip of the complex plane. For $b = \frac{5}{2}$, the strip shrinks to the imaginary axis, but an additional pai…

Orbital stability of peakons for a generalized Camassa-Holm equation with both quadratic and cubic nonlinearity

In this paper, we investigate the orbital stability problem of peakons for a modified Camassa-Holm equation with both quadratic and cubic nonlinearity. This equation was derived from integrable theory and admits peaked soliton (peakon) and multipeakon solutions. By constructing two suitable piecewise functions, we establish the polynomial inequality relating to two conserved quantities and the maximum of the solution to this equation. The error estimate between the maximum of the solution and the peakon then follows from the structure of the polynomial inequality. Finally, we prove that a wave…