Asymptotic stability of solitary waves for the Kaup–Boussinesq system

We consider a Boussinesq type system introduced by Kaup. This model has solitary waves with speeds c ∈ (−1, 1), which are orbitally stable [Angulo, Adv. Differ. Equations 4, 457–492 (1999)]. We prove the asymptotic stability of these solitary waves for c∈(−1,−12), by following the approach for the Korteweg–de Vries equation [Martel and Merle, Math. Ann. 341, 391–427 (2008)] and the Gross-Pitaevskii equation [Béthuel, Gravejat, and Smets, Ann. Sci. Éc. Norm. Supér. 48(4), 1327–1381 (2015)]. The proof relies on a rigidity result around solitary waves, which is proved by using a transformed linearized problem and suitable virial identities. The condition c∈(−1,−12) is only assumed to obtain the positivity of a quadratic form.