This chapter presents the general theory about how different dissipative soliton (DS) dynamics are induced by inhomogeneities and drift. For this purpose, the prototypical Swift-Hohenberg equation (SHE) is used. The chapter focuses on the mechanism that leads to excitable dynamics. It discusses the modeling of fiber cavities in terms of the Lugiato-Lefever model, and describes the effect of periodic pumping in ring cavities. In fiber cavities and microresonators, inhomogeneities and drift are unavoidable due to imperfections in the fabrication process, material properties, and higher-order chromatic light dispersion. In the context of fiber cavities and microresonators, periodic boundary conditions lead to the emission of a train of solitons from the inhomogeneity. These trains of solitons lie at the basis of Kerr frequency comb generation. Therefore, the dynamical instabilities of cavity solitons induced by imperfections of themicroresonator can be particularly relevant for applications relying on stable frequency combs.

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