The effect of a longitudinally periodic group-velocity dispersion (GVD) perturbation on both bright-soliton and platicon comb states is still unclear. We investigate a dual-coupled Kerr microresonator in which GVD varies periodically along the propagation direction while its average is unchanged. The periodic GVD modulates the phase mismatch of opposite sidebands, and the linearized sideband matrix becomes periodic. We use Floquet theory over one modulation period to analyze the modulation instability (MI). In the mixed normal-anomalous $(+/-)$ configuration, the phase-mismatch modulation gives rise to additional MI sidebands at the parametric-resonance condition. Numerical simulations based on the split-step Fourier method show that periodic GVD enhances the breathing of a dissipative Kerr soliton (DKS)–like state. The soliton peak oscillates continuously, and the high-order comb components are redistributed. This breathing state appears in the low-detuning region between the MI-comb and stable-DKS regimes. In the doubly normal $(+/+)$ configuration, the relevant eigenvalue branches remain nearly degenerate in the MI band, and the Floquet sideband channel is strongly suppressed. In the nonlinear regime, periodic GVD reduces the residual ripples on the platicon plateau, while the peak power is almost unchanged.

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• Optical activity
• Optical solitons
• Optical microcavities
• Solitons