C-Integrability Test for Discrete Equations via Multiple Scale Expansions
In this paper we are extending the well known integrability theorems obtained by multiple scale techniques to the case of linearizable difference equations. As an example we apply the theory to the case of a differential-difference dispersive equation of the Burgers hierarchy which via a discrete Ho...
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| Format: | Article |
| Language: | English |
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National Academy of Science of Ukraine
2010-08-01
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| Series: | Symmetry, Integrability and Geometry: Methods and Applications |
| Subjects: | |
| Online Access: | http://dx.doi.org/10.3842/SIGMA.2010.070 |
| _version_ | 1818146977870774272 |
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| author | Christian Scimiterna Decio Levi |
| author_facet | Christian Scimiterna Decio Levi |
| author_sort | Christian Scimiterna |
| collection | DOAJ |
| description | In this paper we are extending the well known integrability theorems obtained by multiple scale techniques to the case of linearizable difference equations. As an example we apply the theory to the case of a differential-difference dispersive equation of the Burgers hierarchy which via a discrete Hopf-Cole transformation reduces to a linear differential difference equation. In this case the equation satisfies the A_1, A_2 and A_3 linearizability conditions. We then consider its discretization. To get a dispersive equation we substitute the time derivative by its symmetric discretization. When we apply to this nonlinear partial difference equation the multiple scale expansion we find out that the lowest order non-secularity condition is given by a non-integrable nonlinear Schrödinger equation. Thus showing that this discretized Burgers equation is neither linearizable not integrable. |
| first_indexed | 2024-12-11T12:27:56Z |
| format | Article |
| id | doaj.art-ec76c9132c61400eb89f4ec89e9fd407 |
| institution | Directory Open Access Journal |
| issn | 1815-0659 |
| language | English |
| last_indexed | 2024-12-11T12:27:56Z |
| publishDate | 2010-08-01 |
| publisher | National Academy of Science of Ukraine |
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| series | Symmetry, Integrability and Geometry: Methods and Applications |
| spelling | doaj.art-ec76c9132c61400eb89f4ec89e9fd4072022-12-22T01:07:19ZengNational Academy of Science of UkraineSymmetry, Integrability and Geometry: Methods and Applications1815-06592010-08-016070C-Integrability Test for Discrete Equations via Multiple Scale ExpansionsChristian ScimiternaDecio LeviIn this paper we are extending the well known integrability theorems obtained by multiple scale techniques to the case of linearizable difference equations. As an example we apply the theory to the case of a differential-difference dispersive equation of the Burgers hierarchy which via a discrete Hopf-Cole transformation reduces to a linear differential difference equation. In this case the equation satisfies the A_1, A_2 and A_3 linearizability conditions. We then consider its discretization. To get a dispersive equation we substitute the time derivative by its symmetric discretization. When we apply to this nonlinear partial difference equation the multiple scale expansion we find out that the lowest order non-secularity condition is given by a non-integrable nonlinear Schrödinger equation. Thus showing that this discretized Burgers equation is neither linearizable not integrable.http://dx.doi.org/10.3842/SIGMA.2010.070linearizable discrete equationslinearizability theoremmultiple scale expansionobstructions to linearizabilitydiscrete Burgers |
| spellingShingle | Christian Scimiterna Decio Levi C-Integrability Test for Discrete Equations via Multiple Scale Expansions Symmetry, Integrability and Geometry: Methods and Applications linearizable discrete equations linearizability theorem multiple scale expansion obstructions to linearizability discrete Burgers |
| title | C-Integrability Test for Discrete Equations via Multiple Scale Expansions |
| title_full | C-Integrability Test for Discrete Equations via Multiple Scale Expansions |
| title_fullStr | C-Integrability Test for Discrete Equations via Multiple Scale Expansions |
| title_full_unstemmed | C-Integrability Test for Discrete Equations via Multiple Scale Expansions |
| title_short | C-Integrability Test for Discrete Equations via Multiple Scale Expansions |
| title_sort | c integrability test for discrete equations via multiple scale expansions |
| topic | linearizable discrete equations linearizability theorem multiple scale expansion obstructions to linearizability discrete Burgers |
| url | http://dx.doi.org/10.3842/SIGMA.2010.070 |
| work_keys_str_mv | AT christianscimiterna cintegrabilitytestfordiscreteequationsviamultiplescaleexpansions AT deciolevi cintegrabilitytestfordiscreteequationsviamultiplescaleexpansions |