abstract: We study the generalized Gross--Neveu and massive Thirring models in 1D. We show that the solitary waves are asymptotic stability of solitary waves in provided that the the linearized operators around them only have zero eigenvalue. We obtain the result by studying the resolvent of the linearized operators and deriving the appropriate linear dispersive decay estimates and Strichartz estimates. This is the joint work with Andrew Comech (Texas A&M University, USA and IITP, Russia) and Atanas Stefanov (University of Kansas, USA).