| This paper mainly studies the multi-peakon solutions of several classes of Camassa-Holm equations and their non-uniqueness in Sobolev space Hs.We first verify that the peakon solutions are weak solutions of the equations,and then study the ODE system that the 2-peakon solutions of the equations need to satisfy.Using differential equations and dynamical system theory,we construct specific 2-peakon solutions and study the collision of 2-peakon solutions,and then use the Fourier transform and time reversibility to prove the non-uniqueness of the 2-peakon solutions for the equations.There are five chapters in this paper.The structure of the article is as follows:In Chapter 1,we mainly introduce the research background,research status and main research contents of the multi-peakon solutions of the Camassa-Holm-type equations and the non-uniqueness of the solutions.In Chapter 2,the multi-peakon solutions and their non-uniqueness of the gmCH equation(l=2)are studied.In chapter 3,the multi-peakon solutions and their non-uniqueness of the gmCH equation(l=3)are considered.In chapter 4,the multi-peakon solutions and their non-uniqueness of the mCH-Novikov equation are discussed.In chapter 5,we summarize this paper and discuss possible directions for future research. |
