Reduction of infinite dimensional equations
In this paper, we use the general Legendre transformation to show the infinite dimensional integrable equations can be reduced to a finite dimensional integrable Hamiltonian system on an invariant set under the flow of the integrable equations. Then we obtain the periodic or quasi-periodic solut...
| Main Authors: | , |
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| Format: | Article |
| Language: | English |
| Published: |
Texas State University
2006-02-01
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| Series: | Electronic Journal of Differential Equations |
| Subjects: | |
| Online Access: | http://ejde.math.txstate.edu/Volumes/2006/17/abstr.html |
| _version_ | 1819206867769884672 |
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| author | Zhongding Li Taixi Xu |
| author_facet | Zhongding Li Taixi Xu |
| author_sort | Zhongding Li |
| collection | DOAJ |
| description | In this paper, we use the general Legendre transformation to show the infinite dimensional integrable equations can be reduced to a finite dimensional integrable Hamiltonian system on an invariant set under the flow of the integrable equations. Then we obtain the periodic or quasi-periodic solution of the equation. This generalizes the results of Lax and Novikov regarding the periodic or quasi-periodic solution of the KdV equation to the general case of isospectral Hamiltonian integrable equation. And finally, we discuss the AKNS hierarchy as a special example. |
| first_indexed | 2024-12-23T05:14:26Z |
| format | Article |
| id | doaj.art-22500218a15c4d1e9b5347a1dcaff10e |
| institution | Directory Open Access Journal |
| issn | 1072-6691 |
| language | English |
| last_indexed | 2024-12-23T05:14:26Z |
| publishDate | 2006-02-01 |
| publisher | Texas State University |
| record_format | Article |
| series | Electronic Journal of Differential Equations |
| spelling | doaj.art-22500218a15c4d1e9b5347a1dcaff10e2022-12-21T17:58:53ZengTexas State UniversityElectronic Journal of Differential Equations1072-66912006-02-01200617115Reduction of infinite dimensional equationsZhongding LiTaixi XuIn this paper, we use the general Legendre transformation to show the infinite dimensional integrable equations can be reduced to a finite dimensional integrable Hamiltonian system on an invariant set under the flow of the integrable equations. Then we obtain the periodic or quasi-periodic solution of the equation. This generalizes the results of Lax and Novikov regarding the periodic or quasi-periodic solution of the KdV equation to the general case of isospectral Hamiltonian integrable equation. And finally, we discuss the AKNS hierarchy as a special example.http://ejde.math.txstate.edu/Volumes/2006/17/abstr.htmlSoliton equationsHamiltonian equationEuler-Lagrange equationintegrable systemsLegendre transformationinvolutive systemsymmetries of equationsinvariant manifoldPoisson bracketsymplectic space. |
| spellingShingle | Zhongding Li Taixi Xu Reduction of infinite dimensional equations Electronic Journal of Differential Equations Soliton equations Hamiltonian equation Euler-Lagrange equation integrable systems Legendre transformation involutive system symmetries of equations invariant manifold Poisson bracket symplectic space. |
| title | Reduction of infinite dimensional equations |
| title_full | Reduction of infinite dimensional equations |
| title_fullStr | Reduction of infinite dimensional equations |
| title_full_unstemmed | Reduction of infinite dimensional equations |
| title_short | Reduction of infinite dimensional equations |
| title_sort | reduction of infinite dimensional equations |
| topic | Soliton equations Hamiltonian equation Euler-Lagrange equation integrable systems Legendre transformation involutive system symmetries of equations invariant manifold Poisson bracket symplectic space. |
| url | http://ejde.math.txstate.edu/Volumes/2006/17/abstr.html |
| work_keys_str_mv | AT zhongdingli reductionofinfinitedimensionalequations AT taixixu reductionofinfinitedimensionalequations |