The modulational instability of a wave in media where the nonlinearity has a focusing-in-space and defocusing-in-time character is investigated. An approximate self-similar solution of the nonlinear Schr\"odinger equation is derived for this nonlinearity and is used as background envelope of the wave. We show that this background wave envelope can initially enhance exponential growth of a transverse spatial modulation. The analysis is extended to the case of a modulation in both time and space, and it is found that such a spatiotemporal modulation can be unstable as well.
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Anderson et al. (1993) studied this question.
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