This paper proves the orbital asymptotic stability of solitary waves for the Amick-Schonbek Boussinesq system, a model for weakly nonlinear long surface waves in shallow water. For initial data sufficiently close to a solitary wave in the weighted Sobolev space H = (H² ∩ L^2, 1) ², the global solution decomposes into a modulated solitary wave plus a remainder U (t) that decays locally to zero at rate t^-1/2 log t in Lⁱnfinity. The modulation speed satisfies uniform bounds |c (t) -c₀| ≤ C ε₀ and |c' (t) | ≤ C ε₀² (1+t) ^-1. Full convergence of c (t) to a limit is not proved; it remains open and is stated as a conditional corollary requiring |c'| in L¹ (e. g. , if |c' (t) | = O ( (1+t) ^-1-ε) for some ε>0). The work confirms the orbital part of Conjecture 1 of Klein-Saut (2024) and provides a complete spectral analysis of the linearized generator Ac = J Lc + c ∂ᵦ, including an Evans function argument, a limiting absorption principle, and sharp t^-1/2 dispersive estimates for the continuous projection Qc.
Maurizio Estefano Mendoza Martínez (Wed,) studied this question.