@article{ba885b3e062744daa244c0fe9a40cb13,
title = "Taming spatiotemporal chaos by impurities in the parametrically driven damped nonlinear Schr{\"o}dinger equation",
abstract = "Solitons of the parametrically driven, damped nonlinear Schr{\"o}dinger equation become unstable and seed spatiotemporal chaos for sufficiently large driving amplitudes. We show that the chaos can be suppressed by introducing localized inhomogeneities in the parameters of the equation. The pinning of the soliton on an {"}attractive{"}inhomogeneity expands its stability region whereas {"}repulsive{"}impurities produce an effective partitioning of the interval. We also show that attractive impurities may spontaneously nucleate solitons which subsequently remain pinned on these defects. A brief account of these results has appeared in patt-sol/9906001, where the interested reader can also find multicolor versions of the figures.",
author = "Alexeeva, {N. V.} and Barashenkov, {I. V.} and Tsironis, {G. P.}",
note = "Funding Information: The analysis of the effect of multiple impurities (as well as the related case of finite intervals) is beyond the scope of this work. Here we only mention that introducing more than one impurity can result in more complicated (though still regular) patterns. We illustrate this by simulating the case of two impurities. As one could expect, when two stable stationary solitons are pinned very far from each other, they do not interact and remain time-independent. However, if the separation is smaller than a certain critical distance, they start exchanging weak radiation waves and develop spontaneous oscillations ... which subsequently synchronize (Fig. 5)! It is important to emphasize that this latter synchronization occurs not among individual elements but among solitons, i.e. clusters of pendula. We have therefore a hierarchy of synchronizations: firstly, the pendula form clusters of synchronous oscillation; secondly, different clusters start oscillating in unison. Acknowledgements. Instructive conversations with Sergei Flach, Vladimir Konotop and Yuri Kosevich, and useful correspondence with Mikhail Bogdan, are gratefully acknowledged. Two of us (NA and IB) wish to cordially thank Nikos Flytzanis for the great time in Crete where this project was started. This research was supported by the NRF of South Africa, Max-Planck-Gesellschaft and the FORTH of Greece. Copyright: Copyright 2012 Elsevier B.V., All rights reserved.",
year = "2001",
month = feb,
doi = "10.2991/jnmp.2001.8.s.2",
language = "English",
volume = "8",
pages = "5--12",
journal = "Journal of Nonlinear Mathematical Physics",
issn = "1402-9251",
publisher = "World Scientific Publishing Co. Pte Ltd",
number = "SUPPL.",
}