Construction of peakon-antipeakon solutions and ill-posedness for the b-family of equations

For s < 3/2, it is known (see [24]) that the Cauchy problem for the b-family of equations is ill-posed in Sobolev spaces H-s when b > 1. The proof of ill-posedness depends naturally on the value of b, and is based on the construction of peakon-antipeakon solutions with interesting properties which allows to make conclusion on ill-posedness. In this context the construction of such type of the solution for b < 1 is very attractive problem which help shed light on the ill-posedness problem in this case. Thus, in the present paper we consider the ill-posedness of the b-family of equations with additional term for insufficiently investigated case b < 1 on the line. (C) 2020 Elsevier Inc. All rights reserved.