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Invertible Darboux Transformations
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- Відділення математики
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- Symmetry, Integrability and Geometry: Methods and Applications (SIGMA)
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- Symmetry, Integrability and Geometry: Methods and Applications, 2013, том 9
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- Symmetry, Integrability and Geometry: Methods and Applications, 2013, том 9, випуск за цей рік
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Invertible Darboux Transformations
Інший ID:
2010 Mathematics Subject Classification: 37K10; 37K15
DOI: http://dx.doi.org/10.3842/SIGMA.2013.002
DOI: http://dx.doi.org/10.3842/SIGMA.2013.002
Посилання:
Invertible Darboux Transformations / E. Shemyakova // Symmetry, Integrability and Geometry: Methods and Applications. — 2013. — Т. 9. — Бібліогр.: 13 назв. — англ.
Підтримка:
This paper is a contribution to the Special Issue “Symmetries of Dif ferential Equations: Frames, Invariants and Applications”. The full collection is available at http://www.emis.de/journals/SIGMA/SDE2012.html.
Дата:
2013
Переглядів:
581
Завантажень:
223
Анотація:
For operators of many different kinds it has been proved that (generalized) Darboux transformations can be built using so called Wronskian formulae. Such Darboux transformations are not invertible in the sense that the corresponding mappings of the operator kernels are not invertible. The only known invertible ones were Laplace transformations (and their compositions), which are special cases of Darboux transformations for hyperbolic bivariate operators of order 2. In the present paper we find a criteria for a bivariate linear partial differential operator of an arbitrary order d to have an invertible Darboux transformation. We show that Wronkian formulae may fail in some cases, and find sufficient conditions for such formulae to work.
