Periodic waves in the discrete mKdV equation: Modulational instability and rogue waves

We derive the traveling periodic waves of the discrete modified Korteweg-de Vries equation by using a nonlinearization method. Modulational stability of the traveling periodic waves is studied from the squared eigenfunction relation and the Lax spectrum. We use numerical approximations to show that, similar to the continuous counterpart, the family of dnoidal solutions is modulationally stable and the family of cnoidal solutions is modulationally unstable. Consequently, algebraic solitons propagate on the dnoidal wave background and rogue waves (spatially and temporally localized events) are dynamically generated on the cnoidal wave background.