Theory of pattern forming systems under traveling-wave forcing

The response of pattern forming systems to external forcing either spatial or temporal has received much attention for several decades. Combined spatio-temporal forcing has only been introduced recently, in particular in the form of a spatially resonant traveling-wave forcing [S. Rüdiger, D.G. Míguez, A.P. Muñuzuri, F. Sagués, J. Casademunt, Phys. Rev. Lett. 90 (2003) 128301]. Since then, both a series of experiments, in the context of Turing patterns in reaction–diffusion systems, and the development of the corresponding generic theory, have unveiled a wealth of new and unexpected phenomena. In this article we review these phenomena, we provide a unified and comprehensive description of them, and extend the theoretical analysis to new situations. We formulate the generic amplitude equations for different orders of spatial resonance for 1d and 2d patterns (stripes and hexagons). We identify and describe in detail the autonomous dynamical system which underlies the phenomenon of traveling-stripe resonance. For 1d we focus on localized solutions (kinks and pulses), their dynamics and their interaction for both 1:1 and 2:1 resonance. Specifically, we discuss the effect of wave-number mismatch (inexact resonance) combined with the non-gradient dynamics induced by the motion of the forcing, resulting in non-trivial interactions and complex spatio-temporal dynamics. Analytical results in the phase approximation are obtained, while numerical techniques are used to study the complete problem. We show that defect interaction is oscillatory with distance, allowing for the existence of locked chaotic wave trains. In 2d we discuss the modulational instabilities of striped patterns and focus mostly on the emergence of hexagons induced by traveling stripe forcing, for exact 1:1 resonance. In this case, we examine in detail the complex bifurcation scenario beyond primary instabilities. We finally discuss the problem of traveling-wave forcing in the context of a specific system, the photosensitive CDIMA reaction, where most experiments have been carried out. We compare experiments and theoretical predictions and propose other experimental systems where the study could be extended. Finally, we review related work by other authors and discuss possible further developments and open questions which hold the promise of new interesting findings.