Local well-posedness and stability of solitary waves for the two-component Dullin–Gottwald–Holm system

In this paper, we study the Cauchy problem for the two-component Dullin–Gottwald–Holm system with the initial data (u0,ρ0)∈Bp,rs(R)×Bp,rs−1(R),1≤p,r≤+∞, and s>max{1+1p,32,2−1p}, which generalizes some previous local well-posed results in Sobolev spaces. Then we prove that all the smooth solitary wave solutions are orbitally stable under small disturbances.