dorsal/arxiv
View Schema$D_4\dot{+} T^2$ Mode Interactions and Hidden Rotational Symmetry
| Authors | John David Crawford |
|---|---|
| Categories | |
| ArXiv ID | patt-sol/9311004 |
| URL | https://arxiv.org/abs/patt-sol/9311004 |
| DOI | 10.1088/0951-7715/7/3/002 |
| Journal | Nonlinearity 7, 697 (1994) |
Abstract
Bifurcation problems in which periodic boundary conditions or Neumann boundary conditions are imposed often involve partial differential equations that have Euclidean symmetry. As a result the normal form equations for the bifurcation may be constrained by the ``hidden'' Euclidean symmetry of the equations, even though this symmetry is broken by the boundary conditions. The effects of such hidden rotation symmetry on $D_4\dot{+} T^2$ mode interactions are studied by analyzing when a $D_4\dot{+} T^2$ symmetric normal form $\tilde{F}$ can be extended to a vector field ${\rm \cal F}$ with Euclidean symmetry. The fundamental case of binary mode interactions between two irreducible representations of $D_4\dot{+} T^2$ is treated in detail. Necessary and sufficient conditions are given that permit $\tilde{F}$ to be extended when the Euclidean group ${\rm \cal E}(2)$ acts irreducibly. When the Euclidean action is reducible, the rotations do not impose any constraints on the normal form of the binary mode interaction. In applications, this dependence on the representation of ${\rm \cal E}(2)$ implies that the effects of hidden rotations are not present if the critical eigenvalues are imaginary. Generalization of these results to more complicated mode interactions is discussed.
{
"annotation_id": "b4e0f0a5-fd72-4947-8d3c-44719a860b57",
"date_created": "2026-03-02T18:00:29.159000Z",
"date_modified": "2026-03-02T18:00:29.159000Z",
"file_hash": "7b0f1d0174b79f2044c6215dac554ed148b7e70f8d480553f24eb8caf57c7da1",
"private": false,
"record": {
"abstract": "Bifurcation problems in which periodic boundary conditions or Neumann\nboundary conditions are imposed often involve partial differential equations\nthat have Euclidean symmetry. As a result the normal form equations for the\nbifurcation may be constrained by the ``hidden\u0027\u0027 Euclidean symmetry of the\nequations, even though this symmetry is broken by the boundary conditions. The\neffects of such hidden rotation symmetry on $D_4\\dot{+} T^2$ mode interactions\nare studied by analyzing when a $D_4\\dot{+} T^2$ symmetric normal form\n$\\tilde{F}$ can be extended to a vector field ${\\rm \\cal F}$ with Euclidean\nsymmetry. The fundamental case of binary mode interactions between two\nirreducible representations of $D_4\\dot{+} T^2$ is treated in detail. Necessary\nand sufficient conditions are given that permit $\\tilde{F}$ to be extended when\nthe Euclidean group ${\\rm \\cal E}(2)$ acts irreducibly. When the Euclidean\naction is reducible, the rotations do not impose any constraints on the normal\nform of the binary mode interaction. In applications, this dependence on the\nrepresentation of ${\\rm \\cal E}(2)$ implies that the effects of hidden\nrotations are not present if the critical eigenvalues are imaginary.\nGeneralization of these results to more complicated mode interactions is\ndiscussed.",
"arxiv_id": "patt-sol/9311004",
"authors": [
"John David Crawford"
],
"categories": [
"patt-sol",
"nlin.PS"
],
"doi": "10.1088/0951-7715/7/3/002",
"journal_ref": "Nonlinearity 7, 697 (1994)",
"title": "$D_4\\dot{+} T^2$ Mode Interactions and Hidden Rotational Symmetry",
"url": "https://arxiv.org/abs/patt-sol/9311004"
},
"schema_id": "dorsal/arxiv",
"source": {
"execution_id": "54dc485a-7bc5-48b3-883d-3c76c02706e4",
"id": "arXiv Dataset IDs",
"type": "Model",
"variant": "snapshot-2026-03-01",
"version": "0.1.0"
},
"user_id": 1000002
}