On the Geometry of Extended Self-Similar Solutions of the Airy Shallow Water Equations
Self-similar solutions of the so-called Airy equations, equivalent to the dispersionless nonlinear Schrödinger equation written in Madelung coordinates, are found and studied from the point of view of complete integrability and of their role in the recurrence relation from a bi-Hamiltonian structure...
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| Опубліковано в: : | Symmetry, Integrability and Geometry: Methods and Applications |
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| Дата: | 2019 |
| Автори: | , , , |
| Формат: | Стаття |
| Мова: | English |
| Опубліковано: |
Інститут математики НАН України
2019
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| Онлайн доступ: | https://nasplib.isofts.kiev.ua/handle/123456789/210301 |
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| Назва журналу: | Digital Library of Periodicals of National Academy of Sciences of Ukraine |
| Цитувати: | On the Geometry of Extended Self-Similar Solutions of the Airy Shallow Water Equations / R. Camassa, G. Falqui, G. Ortenzi, M. Pedroni // Symmetry, Integrability and Geometry: Methods and Applications. — 2019. — Т. 15. — Бібліогр.: 35 назв. — англ. |
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Digital Library of Periodicals of National Academy of Sciences of Ukraine| id |
nasplib_isofts_kiev_ua-123456789-210301 |
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Camassa, R. Falqui, G. Ortenzi, G. Pedroni, M. 2025-12-05T09:27:05Z 2019 On the Geometry of Extended Self-Similar Solutions of the Airy Shallow Water Equations / R. Camassa, G. Falqui, G. Ortenzi, M. Pedroni // Symmetry, Integrability and Geometry: Methods and Applications. — 2019. — Т. 15. — Бібліогр.: 35 назв. — англ. 1815-0659 2010 Mathematics Subject Classification: 37K05; 37J15; 76M55 arXiv: 1907.10920 https://nasplib.isofts.kiev.ua/handle/123456789/210301 https://doi.org/10.3842/SIGMA.2019.087 Self-similar solutions of the so-called Airy equations, equivalent to the dispersionless nonlinear Schrödinger equation written in Madelung coordinates, are found and studied from the point of view of complete integrability and of their role in the recurrence relation from a bi-Hamiltonian structure for the equations. This class of solutions reduces the PDEs to a finite ODE system, which admits several conserved quantities, which allow for to construction of explicit solutions by quadratures and provide the bi-Hamiltonian formulation for the reduced ODEs. RC and MP thank the Dipartimento di Matematica e Applicazioni of Universitá Milano-Bicocca for its hospitality. GF, GO, and MP thank the Carolina Center for Interdisciplinary Applied Mathematics at the University of North Carolina for hosting their visits in 2018. This work was supported by the National Science Foundation under grants RTG DMS-0943851, CMG ARC-1025523, DMS-1009750, DMS-1517879, the Office of Naval Research under grants N00014-18-1-2490 and DURIP N00014-12-1-0749. This project has also received funding under grant H2020-MSCA-RISE-2017 Project No. 778010 IPaDEGAN. All authors gratefully acknowledge the auspices of the GNFM Section of INdAM under which part of this work was carried out. Finally, thanks are also due to the anonymous referees for useful comments and suggestions for further references (e.g., [7, 31, 34]). Their work improved the final form of the present paper. en Інститут математики НАН України Symmetry, Integrability and Geometry: Methods and Applications On the Geometry of Extended Self-Similar Solutions of the Airy Shallow Water Equations Article published earlier |
| institution |
Digital Library of Periodicals of National Academy of Sciences of Ukraine |
| collection |
DSpace DC |
| title |
On the Geometry of Extended Self-Similar Solutions of the Airy Shallow Water Equations |
| spellingShingle |
On the Geometry of Extended Self-Similar Solutions of the Airy Shallow Water Equations Camassa, R. Falqui, G. Ortenzi, G. Pedroni, M. |
| title_short |
On the Geometry of Extended Self-Similar Solutions of the Airy Shallow Water Equations |
| title_full |
On the Geometry of Extended Self-Similar Solutions of the Airy Shallow Water Equations |
| title_fullStr |
On the Geometry of Extended Self-Similar Solutions of the Airy Shallow Water Equations |
| title_full_unstemmed |
On the Geometry of Extended Self-Similar Solutions of the Airy Shallow Water Equations |
| title_sort |
on the geometry of extended self-similar solutions of the airy shallow water equations |
| author |
Camassa, R. Falqui, G. Ortenzi, G. Pedroni, M. |
| author_facet |
Camassa, R. Falqui, G. Ortenzi, G. Pedroni, M. |
| publishDate |
2019 |
| language |
English |
| container_title |
Symmetry, Integrability and Geometry: Methods and Applications |
| publisher |
Інститут математики НАН України |
| format |
Article |
| description |
Self-similar solutions of the so-called Airy equations, equivalent to the dispersionless nonlinear Schrödinger equation written in Madelung coordinates, are found and studied from the point of view of complete integrability and of their role in the recurrence relation from a bi-Hamiltonian structure for the equations. This class of solutions reduces the PDEs to a finite ODE system, which admits several conserved quantities, which allow for to construction of explicit solutions by quadratures and provide the bi-Hamiltonian formulation for the reduced ODEs.
|
| issn |
1815-0659 |
| url |
https://nasplib.isofts.kiev.ua/handle/123456789/210301 |
| citation_txt |
On the Geometry of Extended Self-Similar Solutions of the Airy Shallow Water Equations / R. Camassa, G. Falqui, G. Ortenzi, M. Pedroni // Symmetry, Integrability and Geometry: Methods and Applications. — 2019. — Т. 15. — Бібліогр.: 35 назв. — англ. |
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2025-12-07T21:25:04Z |
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2025-12-07T21:25:04Z |
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