The periodic b -equation and Euler equations on the circle
In this note we show that the periodic b -equation can only be realized as a Euler equation on the Lie group Diff ∞ ( S 1 ) of all smooth and orientation preserving diffeomorphisms on the circle if b = 2 , i.e., for the Camassa–Holm equation. In this case the inertia operator generating the metric o...
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| Veröffentlicht in: | Journal of mathematical physics 2010-05, Vol.51 (5), p.053101-053101-6 |
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| container_end_page | 053101-6 |
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| container_issue | 5 |
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| container_title | Journal of mathematical physics |
| container_volume | 51 |
| creator | Escher, Joachim Seiler, Jörg |
| description | In this note we show that the periodic
b
-equation can only be realized as a Euler equation on the Lie group
Diff
∞
(
S
1
)
of all smooth and orientation preserving diffeomorphisms on the circle if
b
=
2
, i.e., for the Camassa–Holm equation. In this case the inertia operator generating the metric on
Diff
∞
(
S
1
)
is given by
A
=
1
−
∂
x
2
. In contrast, the Degasperis–Procesi equation, for which
b
=
3
, is not a Euler equation on
Diff
∞
(
S
1
)
for any inertia operator. Our result generalizes a recent result of Kolev [“Some geometric investigations on the Degasperis-Procesi shallow water equation,” Wave Motion
46, 412–419 (2009)]. |
| doi_str_mv | 10.1063/1.3405494 |
| format | Article |
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b
-equation can only be realized as a Euler equation on the Lie group
Diff
∞
(
S
1
)
of all smooth and orientation preserving diffeomorphisms on the circle if
b
=
2
, i.e., for the Camassa–Holm equation. In this case the inertia operator generating the metric on
Diff
∞
(
S
1
)
is given by
A
=
1
−
∂
x
2
. In contrast, the Degasperis–Procesi equation, for which
b
=
3
, is not a Euler equation on
Diff
∞
(
S
1
)
for any inertia operator. Our result generalizes a recent result of Kolev [“Some geometric investigations on the Degasperis-Procesi shallow water equation,” Wave Motion
46, 412–419 (2009)].</description><identifier>ISSN: 0022-2488</identifier><identifier>EISSN: 1089-7658</identifier><identifier>DOI: 10.1063/1.3405494</identifier><identifier>CODEN: JMAPAQ</identifier><language>eng</language><publisher>Melville, NY: American Institute of Physics</publisher><subject>Eulers equations ; Exact sciences and technology ; Lie groups ; Mathematical methods in physics ; Mathematical models ; Mathematics ; Nonlinear equations ; Partial differential equations ; Physics ; Sciences and techniques of general use</subject><ispartof>Journal of mathematical physics, 2010-05, Vol.51 (5), p.053101-053101-6</ispartof><rights>American Institute of Physics</rights><rights>2010 American Institute of Physics</rights><rights>2015 INIST-CNRS</rights><rights>Copyright American Institute of Physics May 2010</rights><lds50>peer_reviewed</lds50><woscitedreferencessubscribed>false</woscitedreferencessubscribed><citedby>FETCH-LOGICAL-c471t-c1b57dcdea3de3b541c9c6a1d32c1b036d369c821f89fd4d07f37ac15534af5b3</citedby><cites>FETCH-LOGICAL-c471t-c1b57dcdea3de3b541c9c6a1d32c1b036d369c821f89fd4d07f37ac15534af5b3</cites></display><links><openurl>$$Topenurl_article</openurl><openurlfulltext>$$Topenurlfull_article</openurlfulltext><thumbnail>$$Tsyndetics_thumb_exl</thumbnail><linktohtml>$$Uhttps://pubs.aip.org/jmp/article-lookup/doi/10.1063/1.3405494$$EHTML$$P50$$Gscitation$$H</linktohtml><link.rule.ids>314,776,780,790,1553,4497,27903,27904,76130,76136</link.rule.ids><backlink>$$Uhttp://pascal-francis.inist.fr/vibad/index.php?action=getRecordDetail&idt=22901901$$DView record in Pascal Francis$$Hfree_for_read</backlink></links><search><creatorcontrib>Escher, Joachim</creatorcontrib><creatorcontrib>Seiler, Jörg</creatorcontrib><title>The periodic b -equation and Euler equations on the circle</title><title>Journal of mathematical physics</title><description>In this note we show that the periodic
b
-equation can only be realized as a Euler equation on the Lie group
Diff
∞
(
S
1
)
of all smooth and orientation preserving diffeomorphisms on the circle if
b
=
2
, i.e., for the Camassa–Holm equation. In this case the inertia operator generating the metric on
Diff
∞
(
S
1
)
is given by
A
=
1
−
∂
x
2
. In contrast, the Degasperis–Procesi equation, for which
b
=
3
, is not a Euler equation on
Diff
∞
(
S
1
)
for any inertia operator. Our result generalizes a recent result of Kolev [“Some geometric investigations on the Degasperis-Procesi shallow water equation,” Wave Motion
46, 412–419 (2009)].</description><subject>Eulers equations</subject><subject>Exact sciences and technology</subject><subject>Lie groups</subject><subject>Mathematical methods in physics</subject><subject>Mathematical models</subject><subject>Mathematics</subject><subject>Nonlinear equations</subject><subject>Partial differential equations</subject><subject>Physics</subject><subject>Sciences and techniques of general use</subject><issn>0022-2488</issn><issn>1089-7658</issn><fulltext>true</fulltext><rsrctype>article</rsrctype><creationdate>2010</creationdate><recordtype>article</recordtype><recordid>eNp9kFtLwzAYhoMoOKcX_oMieKHQmS-HNvVOxjzAwJt5HdIcsKO2XdIK_vtlbtMLmRAIfHneNx8PQpeAJ4AzegcTyjBnBTtCI8CiSPOMi2M0wpiQlDAhTtFZCEuMAQRjI3S_eLdJZ33VmkonZZLa1aD6qm0S1ZhkNtTWJ_tRSOK4j7yuvK7tOTpxqg72YneP0dvjbDF9TuevTy_Th3mqWQ59qqHkudHGKmosLTkDXehMgaEkPmGaGZoVWhBwonCGGZw7misNnFOmHC_pGF1tezvfrgYberlsB9_ELyXHuRB5RmiEbraQ9m0I3jrZ-epD-S8JWG7MSJA7M5G93hWqoFXtvGp0FX4ChBQY4omc2HJBV_23gcOlUaPca5SbnW8PRT9b_xuTnXH_wX-XXwPJRJBF</recordid><startdate>20100501</startdate><enddate>20100501</enddate><creator>Escher, Joachim</creator><creator>Seiler, Jörg</creator><general>American Institute of Physics</general><scope>IQODW</scope><scope>AAYXX</scope><scope>CITATION</scope><scope>7U5</scope><scope>8FD</scope><scope>H8D</scope><scope>JQ2</scope><scope>L7M</scope></search><sort><creationdate>20100501</creationdate><title>The periodic b -equation and Euler equations on the circle</title><author>Escher, Joachim ; Seiler, Jörg</author></sort><facets><frbrtype>5</frbrtype><frbrgroupid>cdi_FETCH-LOGICAL-c471t-c1b57dcdea3de3b541c9c6a1d32c1b036d369c821f89fd4d07f37ac15534af5b3</frbrgroupid><rsrctype>articles</rsrctype><prefilter>articles</prefilter><language>eng</language><creationdate>2010</creationdate><topic>Eulers equations</topic><topic>Exact sciences and technology</topic><topic>Lie groups</topic><topic>Mathematical methods in physics</topic><topic>Mathematical models</topic><topic>Mathematics</topic><topic>Nonlinear equations</topic><topic>Partial differential equations</topic><topic>Physics</topic><topic>Sciences and techniques of general use</topic><toplevel>peer_reviewed</toplevel><toplevel>online_resources</toplevel><creatorcontrib>Escher, Joachim</creatorcontrib><creatorcontrib>Seiler, Jörg</creatorcontrib><collection>Pascal-Francis</collection><collection>CrossRef</collection><collection>Solid State and Superconductivity Abstracts</collection><collection>Technology Research Database</collection><collection>Aerospace Database</collection><collection>ProQuest Computer Science Collection</collection><collection>Advanced Technologies Database with Aerospace</collection><jtitle>Journal of mathematical physics</jtitle></facets><delivery><delcategory>Remote Search Resource</delcategory><fulltext>fulltext</fulltext></delivery><addata><au>Escher, Joachim</au><au>Seiler, Jörg</au><format>journal</format><genre>article</genre><ristype>JOUR</ristype><atitle>The periodic b -equation and Euler equations on the circle</atitle><jtitle>Journal of mathematical physics</jtitle><date>2010-05-01</date><risdate>2010</risdate><volume>51</volume><issue>5</issue><spage>053101</spage><epage>053101-6</epage><pages>053101-053101-6</pages><issn>0022-2488</issn><eissn>1089-7658</eissn><coden>JMAPAQ</coden><abstract>In this note we show that the periodic
b
-equation can only be realized as a Euler equation on the Lie group
Diff
∞
(
S
1
)
of all smooth and orientation preserving diffeomorphisms on the circle if
b
=
2
, i.e., for the Camassa–Holm equation. In this case the inertia operator generating the metric on
Diff
∞
(
S
1
)
is given by
A
=
1
−
∂
x
2
. In contrast, the Degasperis–Procesi equation, for which
b
=
3
, is not a Euler equation on
Diff
∞
(
S
1
)
for any inertia operator. Our result generalizes a recent result of Kolev [“Some geometric investigations on the Degasperis-Procesi shallow water equation,” Wave Motion
46, 412–419 (2009)].</abstract><cop>Melville, NY</cop><pub>American Institute of Physics</pub><doi>10.1063/1.3405494</doi><tpages>6</tpages></addata></record> |
| fulltext | fulltext |
| identifier | ISSN: 0022-2488 |
| ispartof | Journal of mathematical physics, 2010-05, Vol.51 (5), p.053101-053101-6 |
| issn | 0022-2488 1089-7658 |
| language | eng |
| recordid | cdi_crossref_primary_10_1063_1_3405494 |
| source | AIP Journals Complete; AIP Digital Archive; Alma/SFX Local Collection |
| subjects | Eulers equations Exact sciences and technology Lie groups Mathematical methods in physics Mathematical models Mathematics Nonlinear equations Partial differential equations Physics Sciences and techniques of general use |
| title | The periodic b -equation and Euler equations on the circle |
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