The interplay of periodic driving and dissipation is a fundamental feature of nonequilibrium physics. We elaborate a scenario for the formation of dissipative multispot excitations (MSEs) in Kerr cavities, modeled by the one- and two-dimensional (1D and 2D) Lugiato-Lefever (LL) equations, which include a spatially periodic pump (SPP). First, we demonstrate that the SPP produces three alternative exact periodic solutions of the LL equation, expressed in terms of the $\mathrm{sn}$, $\mathrm{cn}$, and $\mathrm{dn}$ elliptic functions. By means of numerical methods, we explore the modulational instability (MI) and transverse instability (TI) of the periodic states in 1D and 2D settings, respectively. In the case of the defocusing nonlinearity, the 1D MI breaks the periodic states into an array of spatiotemporal crescents. In the case of self-focusing, the 1D MI, initiated by small random perturbations, leads to the establishment of a chaotic state, with the amplitude statistics featuring a long-tail probability distribution, that represents the presence of dissipative rogue waves. On the other hand, spatially periodic perturbations initiate the formation of breather chains, which periodically disappear and reappear, resembling the Fermi-Pasta-Ulam-Tsingou recurrence. In the 2D regime, the TI results in the formation of an array of 2D lumps. For a given SPP strength, the exact solutions for the periodic structures are stable if the loss constant exceeds a critical value. The findings reported here provide a contribution to nonequilibrium physics in general and may find direct applications in laser physics.

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• Nonlinear optics