The Camassa-Holm equation possesses well-known peaked solitary waves that are called peakons. Their orbital stability has been established by Constantin and Strauss (2000). We prove here the stability of ordered trains of peakons. We also establish a result on the stability of multipeakons

We prove the asymptotic stability in H 1 (R) of the family of solitary waves for the Benjamin-Bona-Mahony equation,We prove that a solution initially close to a solitary wave, once conveniently translated, converges weakly in H 1 (R), as time goes to infinity, to a possibly different solitary wave. The proof is based on a Liouville type theorem for the flow close to the solitary waves, and makes an extensive use of a monotonicity property.

We consider the generalized BBM (Benjamin-Bona-Mahony) equations:2 integer, and the family of solitary wave solutions ϕc(x − x 0 − ct) of this equation. For any p, there exists a necessary and sufficient condition on the speed c > 1 so that a solitary wave solution is nonlinearly stable ([21], [20]). Following the approach of [14] for the generalized KdV equations, we prove that the sum of N sufficiently decoupled stable solitary wave solutions is also stable in the energy space. The proof combines arguments of [21] to prove the stability of a single solitary wave, and monotonicity results of [6]. We also obtain asymptotic stability results following [6]. Using the same tools, we then prove the existence and uniqueness of a solution behaving asymptotically in large time as the sum of N given solitary waves, following the method of [11]. Contents 1. Introduction 402 2. Preliminaries 405 3. Stability proof 412 4. Proof of the asymptotic stability 415 5. Existence and uniqueness of N solitary waves 419 Appendix A. Proof of Lemma 2.3 434 References 436