dorsal/arxiv
View SchemaSources, sinks and wavenumber selection in coupled CGL equations and experimental implications for counter-propagating wave systems
| Authors | Martin van Hecke, Cornelis Storm, Wim van Saarloos |
|---|---|
| Categories | |
| ArXiv ID | patt-sol/9902005 |
| URL | https://arxiv.org/abs/patt-sol/9902005 |
| DOI | 10.1016/S0167-2789(99)00068-8 |
Abstract
We study the coupled complex Ginzburg-Landau (CGL) equations for traveling wave systems in the regime where sources and sinks separate patches of left and right-traveling waves. We show that sources and sinks are the important coherent structures that organize much of the dynamical properties of traveling wave systems. We present in detail the framework to analyze these coherent structures. Our counting arguments for the multiplicities of these structures show that independently of the coefficients of the CGL, there exists a symmetric stationary source solution, which sends out waves with a unique frequency and wave number. Sinks, on the other hand, occur in two-parameter families, and play an essentially passive role. Simulations show that sources can send out stable waves, convectively unstable waves, or absolutely unstable waves. We show that there exists an additional dynamical regime where both single- and bimodal states are unstable; the ensuing chaotic states have no counterpart in single amplitude equations. A third dynamical mechanism is associated with the fact that the width of the sources does not show simple scaling with the growth rate epsilon. In particular, when the group velocity term dominates over the linear growth term, no stationary source can exist, but sources displaying nontrivial dynamics can survive here. Our results are easily accessible by experiments, and we advocate a study of the sources and sinks as a means to probe traveling wave systems and compare theory and experiment.
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"abstract": "We study the coupled complex Ginzburg-Landau (CGL) equations for traveling\nwave systems in the regime where sources and sinks separate patches of left and\nright-traveling waves. We show that sources and sinks are the important\ncoherent structures that organize much of the dynamical properties of traveling\nwave systems. We present in detail the framework to analyze these coherent\nstructures. Our counting arguments for the multiplicities of these structures\nshow that independently of the coefficients of the CGL, there exists a\nsymmetric stationary source solution, which sends out waves with a unique\nfrequency and wave number. Sinks, on the other hand, occur in two-parameter\nfamilies, and play an essentially passive role. Simulations show that sources\ncan send out stable waves, convectively unstable waves, or absolutely unstable\nwaves. We show that there exists an additional dynamical regime where both\nsingle- and bimodal states are unstable; the ensuing chaotic states have no\ncounterpart in single amplitude equations. A third dynamical mechanism is\nassociated with the fact that the width of the sources does not show simple\nscaling with the growth rate epsilon. In particular, when the group velocity\nterm dominates over the linear growth term, no stationary source can exist, but\nsources displaying nontrivial dynamics can survive here. Our results are easily\naccessible by experiments, and we advocate a study of the sources and sinks as\na means to probe traveling wave systems and compare theory and experiment.",
"arxiv_id": "patt-sol/9902005",
"authors": [
"Martin van Hecke",
"Cornelis Storm",
"Wim van Saarloos"
],
"categories": [
"patt-sol",
"nlin.PS"
],
"doi": "10.1016/S0167-2789(99)00068-8",
"title": "Sources, sinks and wavenumber selection in coupled CGL equations and experimental implications for counter-propagating wave systems",
"url": "https://arxiv.org/abs/patt-sol/9902005"
},
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