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In this note, we extend the detailed study of the linearized dynamicsobtained for cnoidal waves of the Korteweg--de Vries equation in \cite{JFA-R}to small-amplitude periodic traveling waves of the generalized Korteweg-deVries equations that are not subject to Benjamin--Feir instability. With theadapted notion of stability, this provides for such waves, global-in-timebounded stability in any Sobolev space, and asymptotic stability of dispersivetype. When doing so, we actually prove that such results also hold for waves ofarbitrary amplitude satisfying a form of spectral stability designated here asdispersive spectral stability.

Linear asymptotic stability of small-amplitude periodic waves of the generalized Korteweg--de Vries equations