dorsal/arxiv
View SchemaDifferential substitutions and symmetries of hyperbolic equations
| Authors | S. Ya. Startsev |
|---|---|
| Categories | |
| ArXiv ID | solv-int/9509006 |
| URL | https://arxiv.org/abs/solv-int/9509006 |
Abstract
There are considered differential substitutions of the form $v=P(x,u,u_{x})$ for which there exists a differential operator $H=\sum^{k}_{i=0} \alpha_{i} D^{i}_{x}$ such that the differential substitution maps the equation $u_{t}=H[s(x,P,D_{x}(P),...,D^{k}_{x}(P))]$ into an evolution equation for any function $s$ and any nonnegative integer $k$. All differential substitutions of the form $v=P(x,u,u_{x})$ known to the author have this property. For example, the well-known Miura transformation $v=u_{x}-u^{2}$ maps any equation of the form $$u_{t}=(D^{2}_{x}+2uD_{x}+2u_{x}) [s(x,u_{x}-u^{2},D_{x}(u_{x}-u^{2}),...,D^{k}_{x}(u_{x}-u^{2}))]$$ into the equation $$v_{t}=(D^{3}_{x}+4vD_{x}+2v_{x})[s(x,v,{{\partial v}\over{\partial x }},...,{{\partial^{k} v}\over{\partial x^{k}}})].$$ The complete classification of such differential substitutions is given. An infinite set of the pairwise nonequivalent differential substitutions with the property mentioned above is constructed. Moreover, a general result about symmetries and invariant functions of hyperbolic equations is obtained.
{
"annotation_id": "8168d9e1-4179-4c4f-886b-d20f46964e65",
"date_created": "2026-03-02T18:02:51.400000Z",
"date_modified": "2026-03-02T18:02:51.400000Z",
"file_hash": "2e35857b88b20162c746f3d0a3b3a566225eccaf4b95aaa15015f176eb80a601",
"private": false,
"record": {
"abstract": "There are considered differential substitutions of the form $v=P(x,u,u_{x})$\nfor which there exists a differential operator $H=\\sum^{k}_{i=0} \\alpha_{i}\nD^{i}_{x}$ such that the differential substitution maps the equation\n$u_{t}=H[s(x,P,D_{x}(P),...,D^{k}_{x}(P))]$ into an evolution equation for any\nfunction $s$ and any nonnegative integer $k$. All differential substitutions of\nthe form $v=P(x,u,u_{x})$ known to the author have this property. For example,\nthe well-known Miura transformation $v=u_{x}-u^{2}$ maps any equation of the\nform $$u_{t}=(D^{2}_{x}+2uD_{x}+2u_{x})\n[s(x,u_{x}-u^{2},D_{x}(u_{x}-u^{2}),...,D^{k}_{x}(u_{x}-u^{2}))]$$ into the\nequation $$v_{t}=(D^{3}_{x}+4vD_{x}+2v_{x})[s(x,v,{{\\partial v}\\over{\\partial x\n}},...,{{\\partial^{k} v}\\over{\\partial x^{k}}})].$$ The complete classification\nof such differential substitutions is given. An infinite set of the pairwise\nnonequivalent differential substitutions with the property mentioned above is\nconstructed. Moreover, a general result about symmetries and invariant\nfunctions of hyperbolic equations is obtained.",
"arxiv_id": "solv-int/9509006",
"authors": [
"S. Ya. Startsev"
],
"categories": [
"solv-int",
"nlin.PS",
"nlin.SI",
"patt-sol"
],
"title": "Differential substitutions and symmetries of hyperbolic equations",
"url": "https://arxiv.org/abs/solv-int/9509006"
},
"schema_id": "dorsal/arxiv",
"source": {
"execution_id": "a5f3e55a-0bbb-4d44-afd1-51e048df4ff9",
"id": "arXiv Dataset IDs",
"type": "Model",
"variant": "snapshot-2026-03-01",
"version": "0.1.0"
},
"user_id": 1000002
}