KdV & KAM:
Gespeichert in:
| 1. Verfasser: | |
|---|---|
| Format: | Elektronisch E-Book |
| Sprache: | English |
| Veröffentlicht: |
Berlin, Heidelberg
Springer Berlin Heidelberg
2003
|
| Ausgabe: | 3. Folge |
| Schriftenreihe: | Ergebnisse der Mathematik und ihrer Grenzgebiete / A Series of Modern Surveys in Mathematics
45 |
| Schlagworte: | |
| Online-Zugang: | Volltext |
| Beschreibung: | In this text the authors consider the Korteweg-de Vries (KdV) equation (ut = - uxxx + 6uux) with periodic boundary conditions. Derived to describe long surface waves in a narrow and shallow channel, this equation in fact models waves in homogeneous, weakly nonlinear and weakly dispersive media in general. Viewing the KdV equation as an infinite dimensional, and in fact integrable Hamiltonian system, we first construct action-angle coordinates which turn out to be globally defined. They make evident that all solutions of the periodic KdV equation are periodic, quasi-periodic or almost-periodic in time. Also, their construction leads to some new results along the way. Subsequently, these coordinates allow us to apply a general KAM theorem for a class of integrable Hamiltonian pde's, proving that large families of periodic and quasi-periodic solutions persist under sufficiently small Hamiltonian perturbations. The pertinent nondegeneracy conditions are verified by calculating the first few Birkhoff normal form terms -- an essentially elementary calculation |
| Beschreibung: | 1 Online-Ressource (XIII, 279 p) |
| ISBN: | 9783662080542 9783642056949 |
| ISSN: | 0071-1136 |
| DOI: | 10.1007/978-3-662-08054-2 |
Internformat
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| 500 | |a In this text the authors consider the Korteweg-de Vries (KdV) equation (ut = - uxxx + 6uux) with periodic boundary conditions. Derived to describe long surface waves in a narrow and shallow channel, this equation in fact models waves in homogeneous, weakly nonlinear and weakly dispersive media in general. Viewing the KdV equation as an infinite dimensional, and in fact integrable Hamiltonian system, we first construct action-angle coordinates which turn out to be globally defined. They make evident that all solutions of the periodic KdV equation are periodic, quasi-periodic or almost-periodic in time. Also, their construction leads to some new results along the way. Subsequently, these coordinates allow us to apply a general KAM theorem for a class of integrable Hamiltonian pde's, proving that large families of periodic and quasi-periodic solutions persist under sufficiently small Hamiltonian perturbations. The pertinent nondegeneracy conditions are verified by calculating the first few Birkhoff normal form terms -- an essentially elementary calculation | ||
| 650 | 4 | |a Mathematics | |
| 650 | 4 | |a Differentiable dynamical systems | |
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Datensatz im Suchindex
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|---|---|
| any_adam_object | |
| author | Kappeler, Thomas |
| author_facet | Kappeler, Thomas |
| author_role | aut |
| author_sort | Kappeler, Thomas |
| author_variant | t k tk |
| building | Verbundindex |
| bvnumber | BV042423404 |
| classification_tum | MAT 000 |
| collection | ZDB-2-SMA ZDB-2-BAE |
| ctrlnum | (OCoLC)1184494459 (DE-599)BVBBV042423404 |
| dewey-full | 514.74 |
| dewey-hundreds | 500 - Natural sciences and mathematics |
| dewey-ones | 514 - Topology |
| dewey-raw | 514.74 |
| dewey-search | 514.74 |
| dewey-sort | 3514.74 |
| dewey-tens | 510 - Mathematics |
| discipline | Mathematik |
| doi_str_mv | 10.1007/978-3-662-08054-2 |
| edition | 3. Folge |
| format | Electronic eBook |
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| id | DE-604.BV042423404 |
| illustrated | Not Illustrated |
| indexdate | 2024-07-10T01:21:13Z |
| institution | BVB |
| isbn | 9783662080542 9783642056949 |
| issn | 0071-1136 |
| language | English |
| oai_aleph_id | oai:aleph.bib-bvb.de:BVB01-027858821 |
| oclc_num | 1184494459 |
| open_access_boolean | |
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| owner_facet | DE-384 DE-703 DE-91 DE-BY-TUM DE-634 |
| physical | 1 Online-Ressource (XIII, 279 p) |
| psigel | ZDB-2-SMA ZDB-2-BAE ZDB-2-SMA_Archive |
| publishDate | 2003 |
| publishDateSearch | 2003 |
| publishDateSort | 2003 |
| publisher | Springer Berlin Heidelberg |
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| series2 | Ergebnisse der Mathematik und ihrer Grenzgebiete / A Series of Modern Surveys in Mathematics |
| spelling | Kappeler, Thomas Verfasser aut KdV & KAM by Thomas Kappeler, Jürgen Pöschel 3. Folge Berlin, Heidelberg Springer Berlin Heidelberg 2003 1 Online-Ressource (XIII, 279 p) txt rdacontent c rdamedia cr rdacarrier Ergebnisse der Mathematik und ihrer Grenzgebiete / A Series of Modern Surveys in Mathematics 45 0071-1136 In this text the authors consider the Korteweg-de Vries (KdV) equation (ut = - uxxx + 6uux) with periodic boundary conditions. Derived to describe long surface waves in a narrow and shallow channel, this equation in fact models waves in homogeneous, weakly nonlinear and weakly dispersive media in general. Viewing the KdV equation as an infinite dimensional, and in fact integrable Hamiltonian system, we first construct action-angle coordinates which turn out to be globally defined. They make evident that all solutions of the periodic KdV equation are periodic, quasi-periodic or almost-periodic in time. Also, their construction leads to some new results along the way. Subsequently, these coordinates allow us to apply a general KAM theorem for a class of integrable Hamiltonian pde's, proving that large families of periodic and quasi-periodic solutions persist under sufficiently small Hamiltonian perturbations. The pertinent nondegeneracy conditions are verified by calculating the first few Birkhoff normal form terms -- an essentially elementary calculation Mathematics Differentiable dynamical systems Global analysis Differential equations, partial Mathematical physics Global Analysis and Analysis on Manifolds Mathematics, general Dynamical Systems and Ergodic Theory Partial Differential Equations Mathematical Methods in Physics Mathematik Mathematische Physik KAM-Theorie (DE-588)4329269-0 gnd rswk-swf Integrables System (DE-588)4114032-1 gnd rswk-swf Partielle Differentialgleichung (DE-588)4044779-0 gnd rswk-swf Normalform (DE-588)4172025-8 gnd rswk-swf Korteweg-de-Vries-Gleichung (DE-588)4287203-0 gnd rswk-swf Korteweg-de-Vries-Gleichung (DE-588)4287203-0 s Integrables System (DE-588)4114032-1 s Normalform (DE-588)4172025-8 s KAM-Theorie (DE-588)4329269-0 s 1\p DE-604 Partielle Differentialgleichung (DE-588)4044779-0 s 2\p DE-604 Pöschel, Jürgen Sonstige oth https://doi.org/10.1007/978-3-662-08054-2 Verlag Volltext 1\p cgwrk 20201028 DE-101 https://d-nb.info/provenance/plan#cgwrk 2\p cgwrk 20201028 DE-101 https://d-nb.info/provenance/plan#cgwrk |
| spellingShingle | Kappeler, Thomas KdV & KAM Mathematics Differentiable dynamical systems Global analysis Differential equations, partial Mathematical physics Global Analysis and Analysis on Manifolds Mathematics, general Dynamical Systems and Ergodic Theory Partial Differential Equations Mathematical Methods in Physics Mathematik Mathematische Physik KAM-Theorie (DE-588)4329269-0 gnd Integrables System (DE-588)4114032-1 gnd Partielle Differentialgleichung (DE-588)4044779-0 gnd Normalform (DE-588)4172025-8 gnd Korteweg-de-Vries-Gleichung (DE-588)4287203-0 gnd |
| subject_GND | (DE-588)4329269-0 (DE-588)4114032-1 (DE-588)4044779-0 (DE-588)4172025-8 (DE-588)4287203-0 |
| title | KdV & KAM |
| title_auth | KdV & KAM |
| title_exact_search | KdV & KAM |
| title_full | KdV & KAM by Thomas Kappeler, Jürgen Pöschel |
| title_fullStr | KdV & KAM by Thomas Kappeler, Jürgen Pöschel |
| title_full_unstemmed | KdV & KAM by Thomas Kappeler, Jürgen Pöschel |
| title_short | KdV & KAM |
| title_sort | kdv kam |
| topic | Mathematics Differentiable dynamical systems Global analysis Differential equations, partial Mathematical physics Global Analysis and Analysis on Manifolds Mathematics, general Dynamical Systems and Ergodic Theory Partial Differential Equations Mathematical Methods in Physics Mathematik Mathematische Physik KAM-Theorie (DE-588)4329269-0 gnd Integrables System (DE-588)4114032-1 gnd Partielle Differentialgleichung (DE-588)4044779-0 gnd Normalform (DE-588)4172025-8 gnd Korteweg-de-Vries-Gleichung (DE-588)4287203-0 gnd |
| topic_facet | Mathematics Differentiable dynamical systems Global analysis Differential equations, partial Mathematical physics Global Analysis and Analysis on Manifolds Mathematics, general Dynamical Systems and Ergodic Theory Partial Differential Equations Mathematical Methods in Physics Mathematik Mathematische Physik KAM-Theorie Integrables System Partielle Differentialgleichung Normalform Korteweg-de-Vries-Gleichung |
| url | https://doi.org/10.1007/978-3-662-08054-2 |
| work_keys_str_mv | AT kappelerthomas kdvkam AT poscheljurgen kdvkam |