dorsal/arxiv
View SchemaWeakly nonlinear analysis in spatially extended systems as a formal perturbation scheme
| Authors | Wolfram Just, Frank Matth"aus, Hans Rainer V"olger, Christine Just, Benno Rumpf, Anja Riegert |
|---|---|
| Categories | |
| ArXiv ID | patt-sol/9706006 |
| URL | https://arxiv.org/abs/patt-sol/9706006 |
Abstract
The well known concept, to reduce the spatio-temporal dynamics beyond instabilities of trivial states to amplitude modulated patterns, is reviewed from the point of view of a formal perturbation expansion for general dissipative partial differential equations. For codimension one instabilities closed analytical formulas for all coefficients of the resulting amplitude equation are given, with no further restriction on the basic equations of motion. Both the autonomous and the explicitly time-dependent case are discussed. For the latter, the problem of strong resonances is addressed separately. The formal character of the expansion allows for an analysis of higher-codimension instabilities like the Turing-Hopf instability and for the discussion of principal limits of the amplitude approach in the present form.
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"date_created": "2026-03-02T18:00:29.282000Z",
"date_modified": "2026-03-02T18:00:29.282000Z",
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"abstract": "The well known concept, to reduce the spatio-temporal dynamics beyond\ninstabilities of trivial states to amplitude modulated patterns, is reviewed\nfrom the point of view of a formal perturbation expansion for general\ndissipative partial differential equations. For codimension one instabilities\nclosed analytical formulas for all coefficients of the resulting amplitude\nequation are given, with no further restriction on the basic equations of\nmotion. Both the autonomous and the explicitly time-dependent case are\ndiscussed. For the latter, the problem of strong resonances is addressed\nseparately. The formal character of the expansion allows for an analysis of\nhigher-codimension instabilities like the Turing-Hopf instability and for the\ndiscussion of principal limits of the amplitude approach in the present form.",
"arxiv_id": "patt-sol/9706006",
"authors": [
"Wolfram Just",
"Frank Matth\"aus",
"Hans Rainer V\"olger",
"Christine Just",
"Benno Rumpf",
"Anja Riegert"
],
"categories": [
"patt-sol",
"nlin.PS"
],
"title": "Weakly nonlinear analysis in spatially extended systems as a formal perturbation scheme",
"url": "https://arxiv.org/abs/patt-sol/9706006"
},
"schema_id": "dorsal/arxiv",
"source": {
"execution_id": "539b39ca-11ab-4590-a441-ee0e17770e4c",
"id": "arXiv Dataset IDs",
"type": "Model",
"variant": "snapshot-2026-03-01",
"version": "0.1.0"
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