dorsal/arxiv
View SchemaPhase Slips and the Eckhaus Instability
| Authors | J. -P. Eckmann, Th. Gallay, C. E. Wayne |
|---|---|
| Categories | |
| ArXiv ID | patt-sol/9503001 |
| URL | https://arxiv.org/abs/patt-sol/9503001 |
| DOI | 10.1088/0951-7715/8/6/004 |
Abstract
We consider the Ginzburg-Landau equation, $ \partial_t u= \partial_x^2 u + u - u|u|^2 $, with complex amplitude $u(x,t)$. We first analyze the phenomenon of phase slips as a consequence of the {\it local} shape of $u$. We next prove a {\it global} theorem about evolution from an Eckhaus unstable state, all the way to the limiting stable finite state, for periodic perturbations of Eckhaus unstable periodic initial data. Equipped with these results, we proceed to prove the corresponding phenomena for the fourth order Swift-Hohenberg equation, of which the Ginzburg-Landau equation is the amplitude approximation. This sheds light on how one should deal with local and global aspects of phase slips for this and many other similar systems.
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"abstract": "We consider the Ginzburg-Landau equation, $ \\partial_t u= \\partial_x^2 u + u\n- u|u|^2 $, with complex amplitude $u(x,t)$. We first analyze the phenomenon of\nphase slips as a consequence of the {\\it local} shape of $u$. We next prove a\n{\\it global} theorem about evolution from an Eckhaus unstable state, all the\nway to the limiting stable finite state, for periodic perturbations of Eckhaus\nunstable periodic initial data. Equipped with these results, we proceed to\nprove the corresponding phenomena for the fourth order Swift-Hohenberg\nequation, of which the Ginzburg-Landau equation is the amplitude approximation.\nThis sheds light on how one should deal with local and global aspects of phase\nslips for this and many other similar systems.",
"arxiv_id": "patt-sol/9503001",
"authors": [
"J. -P. Eckmann",
"Th. Gallay",
"C. E. Wayne"
],
"categories": [
"patt-sol",
"nlin.PS"
],
"doi": "10.1088/0951-7715/8/6/004",
"title": "Phase Slips and the Eckhaus Instability",
"url": "https://arxiv.org/abs/patt-sol/9503001"
},
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