On the Hill Discriminant of Lamé's Differential Equation
Lamé's differential equation is a linear differential equation of the second order with a periodic coefficient involving the Jacobian elliptic function sn, depending on the modulus , and two additional parameters and . This differential equation appears in several applications, for example, th...
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| Veröffentlicht in: | Symmetry, Integrability and Geometry: Methods and Applications |
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| Datum: | 2024 |
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| Format: | Artikel |
| Sprache: | Englisch |
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Інститут математики НАН України
2024
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| Online Zugang: | https://nasplib.isofts.kiev.ua/handle/123456789/212174 |
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| Назва журналу: | Digital Library of Periodicals of National Academy of Sciences of Ukraine |
| Zitieren: | On the Hill Discriminant of Lamé's Differential Equation. Hans Volkmer. SIGMA 20 (2024), 021, 9 pages |
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Digital Library of Periodicals of National Academy of Sciences of Ukraine| _version_ | 1862742066666143744 |
|---|---|
| author | Volkmer, Hans |
| author_facet | Volkmer, Hans |
| citation_txt | On the Hill Discriminant of Lamé's Differential Equation. Hans Volkmer. SIGMA 20 (2024), 021, 9 pages |
| collection | DSpace DC |
| container_title | Symmetry, Integrability and Geometry: Methods and Applications |
| description | Lamé's differential equation is a linear differential equation of the second order with a periodic coefficient involving the Jacobian elliptic function sn, depending on the modulus , and two additional parameters and . This differential equation appears in several applications, for example, the motion of coupled particles in a periodic potential. Stability and existence of periodic solutions of Lamé's equations are determined by the value of its Hill discriminant (, , ). The Hill discriminant is compared to an explicitly known quantity, including explicit error bounds. This result is derived from the observation that Lamé's equation with = 1 can be solved by hypergeometric functions because then the elliptic function reduces to the hyperbolic tangent function. A connection relation between hypergeometric functions then allows the approximation of the Hill discriminant by a simple expression. In particular, one obtains an asymptotic approximation of (, , ) when the modulus tends to 1.
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| first_indexed | 2026-03-21T18:11:27Z |
| format | Article |
| fulltext | |
| id | nasplib_isofts_kiev_ua-123456789-212174 |
| institution | Digital Library of Periodicals of National Academy of Sciences of Ukraine |
| issn | 1815-0659 |
| language | English |
| last_indexed | 2026-03-21T18:11:27Z |
| publishDate | 2024 |
| publisher | Інститут математики НАН України |
| record_format | dspace |
| spelling | Volkmer, Hans 2026-01-30T08:19:23Z 2024 On the Hill Discriminant of Lamé's Differential Equation. Hans Volkmer. SIGMA 20 (2024), 021, 9 pages 1815-0659 2020 Mathematics Subject Classification: 33E10; 34D20 arXiv:2306.12539 https://nasplib.isofts.kiev.ua/handle/123456789/212174 https://doi.org/10.3842/SIGMA.2024.021 Lamé's differential equation is a linear differential equation of the second order with a periodic coefficient involving the Jacobian elliptic function sn, depending on the modulus , and two additional parameters and . This differential equation appears in several applications, for example, the motion of coupled particles in a periodic potential. Stability and existence of periodic solutions of Lamé's equations are determined by the value of its Hill discriminant (, , ). The Hill discriminant is compared to an explicitly known quantity, including explicit error bounds. This result is derived from the observation that Lamé's equation with = 1 can be solved by hypergeometric functions because then the elliptic function reduces to the hyperbolic tangent function. A connection relation between hypergeometric functions then allows the approximation of the Hill discriminant by a simple expression. In particular, one obtains an asymptotic approximation of (, , ) when the modulus tends to 1. The author thanks the anonymous referees whose remarks led to an improvement of the paper. en Інститут математики НАН України Symmetry, Integrability and Geometry: Methods and Applications On the Hill Discriminant of Lamé's Differential Equation Article published earlier |
| spellingShingle | On the Hill Discriminant of Lamé's Differential Equation Volkmer, Hans |
| title | On the Hill Discriminant of Lamé's Differential Equation |
| title_full | On the Hill Discriminant of Lamé's Differential Equation |
| title_fullStr | On the Hill Discriminant of Lamé's Differential Equation |
| title_full_unstemmed | On the Hill Discriminant of Lamé's Differential Equation |
| title_short | On the Hill Discriminant of Lamé's Differential Equation |
| title_sort | on the hill discriminant of lamé's differential equation |
| url | https://nasplib.isofts.kiev.ua/handle/123456789/212174 |
| work_keys_str_mv | AT volkmerhans onthehilldiscriminantoflamesdifferentialequation |