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Abstract Cavity solitons, which are dissipative solitons with a finite extension that appear in the transverse plane of nonlinear optical cavities, have been advocated for use in fast and compact optical information storage. We discuss the instabilities that can affect cavity solitons appearing in Kerr cavities. In particular, cavity solitons may exhibit a Hopf bifurcation leading to self-pulsating behavior, which is then followed by the destruction of the oscillation in a saddle-loop bifurcation. Beyond this point, there is a regime of excitable cavity solitons which appear when suitable perturbations are applied. Excitability is characterized by the nonlinear response of the system upon the application of an external stimulus. Only stimuli exceeding a threshold value are able to elicit a full and well-defined response in the system. In the case of cavity solitons, excitability emerges from the spatial dependence, since the system does not exhibit any excitable behavior locally. We demonstrate the existence of two different mechanisms which lead to excitability, depending on the profile of the pump field.

P. Colet IFISC, Instituto de F´ısica Interdisciplinar y Sistemas Complejos, (CSIC-UIB), Campus Universitat de les Illes Balears, E-07122 Palma de Mallorca, Spain, [email protected] D. Gomila IFISC, Instituto de F´ısica Interdisciplinar y Sistemas Complejos, (CSIC-UIB), Campus Universitat de les Illes Balears, E-07122 Palma de Mallorca, Spain, [email protected] A. Jacobo IFISC, Instituto de F´ısica Interdisciplinar y Sistemas Complejos, (CSIC-UIB), Campus Universitat de les Illes Balears, E-07122 Palma de Mallorca, Spain, [email protected] M.A. Mat´ıas IFISC, Instituto de F´ısica Interdisciplinar y Sistemas Complejos, (CSIC-UIB), Campus Universitat de les Illes Balears, E-07122 Palma de Mallorca, Spain, [email protected] Colet, P. et al.: Excitability Mediated by Dissipative Solitons in Nonlinear Optical Cavities. Lect. Notes Phys. 751, 113–135 (2008) c Springer-Verlag Berlin Heidelberg 2008 DOI 10.1007/978-3-540-78217-9 5 

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1 Introduction The concept of excitability was initially introduced in the context of biological systems, e.g., to describe neuron firing, and it has been found to be present in a wide variety of systems [1, 2], including optical systems [3, 4, 5, 6, 7]. Typically, a system is considered to be excitable if the response of the system to perturbations of the stationary state varies greatly, depending on whether the amplitude of the perturbation exceeds a threshold value. Thus, while small perturbations induce a smooth return to the fixed point, above-threshold perturbations induce a large phase space excursion (firing) before coming back to the rest state. Furthermore, after one firing, the system cannot be excited again within a refractory period of time. In phase space [8, 9], excitability occurs for parameter regions where a stable fixed point is close to a bifurcation in which an oscillation is created. A well-known example of an excitable system is the FitzHugh–N