Abstract
<abstract> In this paper, we consider the existence of periodic and kink wave solutions of a perturbed defocusing mKdV equation. Based on geometric singular perturbation theory, Chebyshev criteria and bifurcation theory of dynamic system, the wave speed conditions for the periodic and kink solutions are given. The monotonicity of the wave speed is proved, and moreover the upper and lower bounds of the limiting wave speeds are obtained. The uniqueness of the periodic waves is established by showing that the Abelian integrals form a Chebyshev set. In addition, there is no coexistence of one periodic and one solitary waves. The proof process does not need any explicit expression of the original defocusing mKdV periodic wave or kink wave solutions. </abstract>
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