Stability of smooth solitary waves in the \(b\)-Camassa-Holm equation (Q2083725)

This paper deals with stability of smooth solitary waves of the \(b\)-family of equations with non-zero background. The notion of orbital stability used for the authors is presented in Definition 1, while the main result is given in Theorem 1. An important highlight is that the class of solutions under consideration is not exactly functions in the class of Sobolev spaces, as usually required in works dealing with well-posedness of this sort of equations, but actually, an affine Sobolev space, as clearly shown in the space of functions defined in [\textit{D. D. Holm} and \textit{M. F. Staley}, SIAM J. Appl. Dyn. Syst. 2, No. 3, 323--380 (2003; Zbl 1088.76531)]. The main stability result is numerically verified for values \(b>1\) for two different limits for the background, with exception of \(b=2\) and \(b=3\), which correspond to the well known Camassa-Holm and Degasperis-Procesi equations.