The dynamics of spatiotemporal modulations

The modulational instability of traveling waves is often thought to be a crucial point in the mechanism of transition to space-time disorder and turbulence. The aim of this paper is to study the effect of spatiotemporal modulations on some dynamics u(0)(x,t), which may occur as an instability process when a control parameter varies, for instance. We analyze the properties of the modulated dynamics of the form g(1)(x)g(2)(t)u(0)(x,t) compared to those of the reference dynamics u(0)(x,t), using operator theory. We show that, if the reference dynamics is invariant under some space-time symmetry in the sense of Ref. J. Nonlinear Sci. 2, 183 (1992)], the modulation has the effect of either deforming this symmetry or breaking it, depending on whether the corresponding operator remains unitary or not. We also demonstrate that the smallest Euclidean space containing the modulated dynamics has a dimension smaller than or equal to the smallest Euclidean space containing u(0)(x,t). The previous results are then applied to the case of modulated uniformly traveling waves. While the spatiotemporal translation invariance of the wave never persists in the presence of a modulation, the existence of a spatiotemporal symmetry depends on the resonance of the Fourier sidebands due to the modulation. In case of nonresonance, a spatiotemporal symmetry exists and is explicitly determined. In this situation, the modulated wave and the carrier wave have the same spectrum (up to a normalization factor), the same entropy, and the spatial (resp., temporal) two-point correlation is deformed only by the spatial (resp., temporal) modulation. (c) 1995 American Institute of Physics.