This paper mainly studies the explicit wave-breaking mechanism and dynamical behavior of solutions near the explicit self-similar singularity for the Camassa-Holm and Degasperis-Procesi equations, which can be regarded as a model for shallow water dynamics and arising from the approximation of the Hamiltonian for Euler's equation in the shallow water regime. We prove that the Camassa-Holm and Degasperis-Procesi equations admit stable explicit self-similar solutions. After that, the nonlinear instability of explicit self-similar solution for the Korteweg-de Vries equation is given.
"On the stable self-similar waves for the Camassa-Holm and Degasperis-Procesi equations." Adv. Differential Equations 25 (5/6) 315 - 334, May/June 2020.