dorsal/arxiv
View SchemaProperties of equations of the continuous Toda type
| Authors | E. Alfinito, G. Profilo, G. Soliani |
|---|---|
| Categories | |
| ArXiv ID | solv-int/9511008 |
| URL | https://arxiv.org/abs/solv-int/9511008 |
| DOI | 10.1088/0305-4470/30/5/019 |
| Journal | J.Phys.A30:1527-1547,1997 |
Abstract
We study a modified version of an equation of the continuous Toda type in 1+1 dimensions. This equation contains a friction-like term which can be switched off by annihilating a free parameter $\ep$. We apply the prolongation method, the symmetry and the approximate symmetry approach. This strategy allows us to get insight into both the equations for $\ep =0$ and $\ep \ne 0$, whose properties arising in the above frameworks are mutually compared. For $\ep =0$, the related prolongation equations are solved by means of certain series expansions which lead to an infinite- dimensional Lie algebra. Furthermore, using a realization of the Lie algebra of the Euclidean group $E_{2}$, a connection is shown between the continuous Toda equation and a linear wave equation which resembles a special case of a three-dimensional wave equation that occurs in a generalized Gibbons-Hawking ansatz \cite{lebrun}. Nontrivial solutions to the wave equation expressed in terms of Bessel functions are determined. For $\ep\,\ne\,0,$ we obtain a finite-dimensional Lie algebra with four elements. A matrix representation of this algebra yields solutions of the modified continuous Toda equation associated with a reduced form of a perturbative Liouville equation. This result coincides with that achieved in the context of the approximate symmetry approach. Example of exact solutions are also provided. In particular, the inverse of the exponential-integral function turns out to be defined by the reduced differential equation coming from a linear combination of the time and space translations. Finally, a Lie algebra characterizing the approximate symmetries is discussed.
{
"annotation_id": "63d5d722-f851-4333-944b-3159ad0ddf1d",
"date_created": "2026-03-02T18:02:50.573000Z",
"date_modified": "2026-03-02T18:02:50.573000Z",
"file_hash": "a07dc24b94f65802fc4d79c40c8eb302b39d93fbe286a6c62ff48f491d1bbf3d",
"private": false,
"record": {
"abstract": "We study a modified version of an equation of the continuous Toda type in 1+1\ndimensions. This equation contains a friction-like term which can be switched\noff by annihilating a free parameter $\\ep$. We apply the prolongation method,\nthe symmetry and the approximate symmetry approach. This strategy allows us to\nget insight into both the equations for $\\ep =0$ and $\\ep \\ne 0$, whose\nproperties arising in the above frameworks are mutually compared. For $\\ep =0$,\nthe related prolongation equations are solved by means of certain series\nexpansions which lead to an infinite- dimensional Lie algebra. Furthermore,\nusing a realization of the Lie algebra of the Euclidean group $E_{2}$, a\nconnection is shown between the continuous Toda equation and a linear wave\nequation which resembles a special case of a three-dimensional wave equation\nthat occurs in a generalized Gibbons-Hawking ansatz \\cite{lebrun}. Nontrivial\nsolutions to the wave equation expressed in terms of Bessel functions are\ndetermined.\n For $\\ep\\,\\ne\\,0,$ we obtain a finite-dimensional Lie algebra with four\nelements. A matrix representation of this algebra yields solutions of the\nmodified continuous Toda equation associated with a reduced form of a\nperturbative Liouville equation. This result coincides with that achieved in\nthe context of the approximate symmetry approach. Example of exact solutions\nare also provided. In particular, the inverse of the exponential-integral\nfunction turns out to be defined by the reduced differential equation coming\nfrom a linear combination of the time and space translations. Finally, a Lie\nalgebra characterizing the approximate symmetries is discussed.",
"arxiv_id": "solv-int/9511008",
"authors": [
"E. Alfinito",
"G. Profilo",
"G. Soliani"
],
"categories": [
"solv-int",
"gr-qc",
"hep-th",
"nlin.SI"
],
"doi": "10.1088/0305-4470/30/5/019",
"journal_ref": "J.Phys.A30:1527-1547,1997",
"title": "Properties of equations of the continuous Toda type",
"url": "https://arxiv.org/abs/solv-int/9511008"
},
"schema_id": "dorsal/arxiv",
"source": {
"execution_id": "7b49a924-7d91-49f9-9bd9-4b41af518a00",
"id": "arXiv Dataset IDs",
"type": "Model",
"variant": "snapshot-2026-03-01",
"version": "0.1.0"
},
"user_id": 1000002
}