dorsal/arxiv
View SchemaDiffusive Mixing of Stable States in the Ginzburg-Landau Equation
| Authors | Thierry Gallay, Alexander Mielke |
|---|---|
| Categories | |
| ArXiv ID | patt-sol/9801003 |
| URL | https://arxiv.org/abs/patt-sol/9801003 |
| DOI | 10.1007/s002200050495 |
Abstract
For the time-dependent Ginzburg-Landau equation on the real line, we construct solutions which converge, as $x \to \pm\infty$, to periodic stationary states with different wave-numbers $\eta_\pm$. These solutions are stable with respect to small perturbations, and approach as $t \to +\infty$ a universal diffusive profile depending only on the values of $\eta_\pm$. This extends a previous result of Bricmont and Kupiainen by removing the assumption that $\eta_\pm$ should be close to zero. The existence of the diffusive profile is obtained as an application of the theory of monotone operators, and the long-time behavior of our solutions is controlled by rewriting the system in scaling variables and using energy estimates involving an exponentially growing damping term.
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"abstract": "For the time-dependent Ginzburg-Landau equation on the real line, we\nconstruct solutions which converge, as $x \\to \\pm\\infty$, to periodic\nstationary states with different wave-numbers $\\eta_\\pm$. These solutions are\nstable with respect to small perturbations, and approach as $t \\to +\\infty$ a\nuniversal diffusive profile depending only on the values of $\\eta_\\pm$. This\nextends a previous result of Bricmont and Kupiainen by removing the assumption\nthat $\\eta_\\pm$ should be close to zero. The existence of the diffusive profile\nis obtained as an application of the theory of monotone operators, and the\nlong-time behavior of our solutions is controlled by rewriting the system in\nscaling variables and using energy estimates involving an exponentially growing\ndamping term.",
"arxiv_id": "patt-sol/9801003",
"authors": [
"Thierry Gallay",
"Alexander Mielke"
],
"categories": [
"patt-sol",
"nlin.PS"
],
"doi": "10.1007/s002200050495",
"title": "Diffusive Mixing of Stable States in the Ginzburg-Landau Equation",
"url": "https://arxiv.org/abs/patt-sol/9801003"
},
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