Stability of peakons and periodic peakons for a nonlinear quartic Camassa-Holm equation

In this paper, we study the orbital stability of peakons and periodic peakons for a nonlinear quartic Camassa-Holm equation (QCHE).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

Orbital stability of smooth solitary waves for the bb-family of Camassa-Holm equations

In this paper, we study the stability of smooth solitary waves for the $b$-family of Camassa-Holm equations. We verify the stability criterion analytically for the general case $b>1$ by the idea of the monotonicity of the period function for planar Hamiltonian systems and show that the smooth solitary waves are orbitally stable, which gives a positive answer to the open problem proposed by Lafortune and Pelinovsky [S. Lafortune, D. E. Pelinovsky, Stability of smooth solitary waves in the $b$-Camassa-Holm equation]