Quasi-periodic Solutions for a Generalized Higher-Order Boussinesq Equation
In this paper one-dimensional generalized eighth-order Boussinesq equation u tt - ∂ x 2 u + β ∂ x 4 u - ∂ x 6 u + ∂ x 8 u + ( u 3 ) xx = 0 , β = ± 1 with the boundary conditions u ( 0 , t ) = u ( π , t ) = u xx ( 0 , t ) = u xx ( π , t ) = u xxxx ( 0 , t ) = u xxxx ( π , t ) = 0 is considered. It is...
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| Veröffentlicht in: | Qualitative theory of dynamical systems 2023-12, Vol.22 (4), Article 139 |
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| creator | Shi, Yanling Xu, Junxiang |
| description | In this paper one-dimensional generalized eighth-order Boussinesq equation
u
tt
-
∂
x
2
u
+
β
∂
x
4
u
-
∂
x
6
u
+
∂
x
8
u
+
(
u
3
)
xx
=
0
,
β
=
±
1
with the boundary conditions
u
(
0
,
t
)
=
u
(
π
,
t
)
=
u
xx
(
0
,
t
)
=
u
xx
(
π
,
t
)
=
u
xxxx
(
0
,
t
)
=
u
xxxx
(
π
,
t
)
=
0
is considered. It is proved that the above equation admits small-amplitude quasi-periodic solutions. The proof is based on an infinite dimensional KAM theorem and Birkhoff normal form. |
| doi_str_mv | 10.1007/s12346-023-00840-w |
| format | Article |
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u
tt
-
∂
x
2
u
+
β
∂
x
4
u
-
∂
x
6
u
+
∂
x
8
u
+
(
u
3
)
xx
=
0
,
β
=
±
1
with the boundary conditions
u
(
0
,
t
)
=
u
(
π
,
t
)
=
u
xx
(
0
,
t
)
=
u
xx
(
π
,
t
)
=
u
xxxx
(
0
,
t
)
=
u
xxxx
(
π
,
t
)
=
0
is considered. It is proved that the above equation admits small-amplitude quasi-periodic solutions. The proof is based on an infinite dimensional KAM theorem and Birkhoff normal form.</description><identifier>ISSN: 1575-5460</identifier><identifier>EISSN: 1662-3592</identifier><identifier>DOI: 10.1007/s12346-023-00840-w</identifier><language>eng</language><publisher>Cham: Springer International Publishing</publisher><subject>Boundary conditions ; Boussinesq equations ; Canonical forms ; Difference and Functional Equations ; Dynamical Systems and Ergodic Theory ; Mathematics ; Mathematics and Statistics</subject><ispartof>Qualitative theory of dynamical systems, 2023-12, Vol.22 (4), Article 139</ispartof><rights>The Author(s), under exclusive licence to Springer Nature Switzerland AG 2023. Springer Nature or its licensor (e.g. a society or other partner) holds exclusive rights to this article under a publishing agreement with the author(s) or other rightsholder(s); author self-archiving of the accepted manuscript version of this article is solely governed by the terms of such publishing agreement and applicable law.</rights><lds50>peer_reviewed</lds50><woscitedreferencessubscribed>false</woscitedreferencessubscribed><citedby>FETCH-LOGICAL-c319t-e1b4af2ac0aa57176f587ab57c58879fa86ddefc7d07ab6d98b0d5abec201f043</citedby><cites>FETCH-LOGICAL-c319t-e1b4af2ac0aa57176f587ab57c58879fa86ddefc7d07ab6d98b0d5abec201f043</cites></display><links><openurl>$$Topenurl_article</openurl><openurlfulltext>$$Topenurlfull_article</openurlfulltext><thumbnail>$$Tsyndetics_thumb_exl</thumbnail><linktopdf>$$Uhttps://link.springer.com/content/pdf/10.1007/s12346-023-00840-w$$EPDF$$P50$$Gspringer$$H</linktopdf><linktohtml>$$Uhttps://link.springer.com/10.1007/s12346-023-00840-w$$EHTML$$P50$$Gspringer$$H</linktohtml><link.rule.ids>314,777,781,27905,27906,41469,42538,51300</link.rule.ids></links><search><creatorcontrib>Shi, Yanling</creatorcontrib><creatorcontrib>Xu, Junxiang</creatorcontrib><title>Quasi-periodic Solutions for a Generalized Higher-Order Boussinesq Equation</title><title>Qualitative theory of dynamical systems</title><addtitle>Qual. Theory Dyn. Syst</addtitle><description>In this paper one-dimensional generalized eighth-order Boussinesq equation
u
tt
-
∂
x
2
u
+
β
∂
x
4
u
-
∂
x
6
u
+
∂
x
8
u
+
(
u
3
)
xx
=
0
,
β
=
±
1
with the boundary conditions
u
(
0
,
t
)
=
u
(
π
,
t
)
=
u
xx
(
0
,
t
)
=
u
xx
(
π
,
t
)
=
u
xxxx
(
0
,
t
)
=
u
xxxx
(
π
,
t
)
=
0
is considered. It is proved that the above equation admits small-amplitude quasi-periodic solutions. The proof is based on an infinite dimensional KAM theorem and Birkhoff normal form.</description><subject>Boundary conditions</subject><subject>Boussinesq equations</subject><subject>Canonical forms</subject><subject>Difference and Functional Equations</subject><subject>Dynamical Systems and Ergodic Theory</subject><subject>Mathematics</subject><subject>Mathematics and Statistics</subject><issn>1575-5460</issn><issn>1662-3592</issn><fulltext>true</fulltext><rsrctype>article</rsrctype><creationdate>2023</creationdate><recordtype>article</recordtype><recordid>eNp9kE1LAzEQhoMoWKt_wNOC5-gkm4_do5baikIR9Ryym6Sm1E2bdCn6601dwZunGYb3eQcehC4JXBMAeZMILZnAQEsMUDHA-yM0IkJQXPKaHuedS445E3CKzlJaAQgqSzpCj8-9Th5vbPTB-LZ4Cet-50OXChdioYuZ7WzUa_9lTTH3y3cb8SIaG4u70KfkO5u2xXTb6wNzjk6cXid78TvH6O1--jqZ46fF7GFy-4TbktQ7bEnDtKO6Ba25JFI4XkndcNnyqpK105UwxrpWGshnYeqqAcN1Y1sKxAErx-hq6N3EsO1t2qlV6GOXXypaMcoZ46zOKTqk2hhSitapTfQfOn4qAuogTQ3SVJamfqSpfYbKAUo53C1t_Kv-h_oGr9Vw9w</recordid><startdate>20231201</startdate><enddate>20231201</enddate><creator>Shi, Yanling</creator><creator>Xu, Junxiang</creator><general>Springer International Publishing</general><general>Springer Nature B.V</general><scope>AAYXX</scope><scope>CITATION</scope></search><sort><creationdate>20231201</creationdate><title>Quasi-periodic Solutions for a Generalized Higher-Order Boussinesq Equation</title><author>Shi, Yanling ; Xu, Junxiang</author></sort><facets><frbrtype>5</frbrtype><frbrgroupid>cdi_FETCH-LOGICAL-c319t-e1b4af2ac0aa57176f587ab57c58879fa86ddefc7d07ab6d98b0d5abec201f043</frbrgroupid><rsrctype>articles</rsrctype><prefilter>articles</prefilter><language>eng</language><creationdate>2023</creationdate><topic>Boundary conditions</topic><topic>Boussinesq equations</topic><topic>Canonical forms</topic><topic>Difference and Functional Equations</topic><topic>Dynamical Systems and Ergodic Theory</topic><topic>Mathematics</topic><topic>Mathematics and Statistics</topic><toplevel>peer_reviewed</toplevel><toplevel>online_resources</toplevel><creatorcontrib>Shi, Yanling</creatorcontrib><creatorcontrib>Xu, Junxiang</creatorcontrib><collection>CrossRef</collection><jtitle>Qualitative theory of dynamical systems</jtitle></facets><delivery><delcategory>Remote Search Resource</delcategory><fulltext>fulltext</fulltext></delivery><addata><au>Shi, Yanling</au><au>Xu, Junxiang</au><format>journal</format><genre>article</genre><ristype>JOUR</ristype><atitle>Quasi-periodic Solutions for a Generalized Higher-Order Boussinesq Equation</atitle><jtitle>Qualitative theory of dynamical systems</jtitle><stitle>Qual. Theory Dyn. Syst</stitle><date>2023-12-01</date><risdate>2023</risdate><volume>22</volume><issue>4</issue><artnum>139</artnum><issn>1575-5460</issn><eissn>1662-3592</eissn><abstract>In this paper one-dimensional generalized eighth-order Boussinesq equation
u
tt
-
∂
x
2
u
+
β
∂
x
4
u
-
∂
x
6
u
+
∂
x
8
u
+
(
u
3
)
xx
=
0
,
β
=
±
1
with the boundary conditions
u
(
0
,
t
)
=
u
(
π
,
t
)
=
u
xx
(
0
,
t
)
=
u
xx
(
π
,
t
)
=
u
xxxx
(
0
,
t
)
=
u
xxxx
(
π
,
t
)
=
0
is considered. It is proved that the above equation admits small-amplitude quasi-periodic solutions. The proof is based on an infinite dimensional KAM theorem and Birkhoff normal form.</abstract><cop>Cham</cop><pub>Springer International Publishing</pub><doi>10.1007/s12346-023-00840-w</doi></addata></record> |
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| ispartof | Qualitative theory of dynamical systems, 2023-12, Vol.22 (4), Article 139 |
| issn | 1575-5460 1662-3592 |
| language | eng |
| recordid | cdi_proquest_journals_2842544549 |
| source | SpringerLink Journals - AutoHoldings |
| subjects | Boundary conditions Boussinesq equations Canonical forms Difference and Functional Equations Dynamical Systems and Ergodic Theory Mathematics Mathematics and Statistics |
| title | Quasi-periodic Solutions for a Generalized Higher-Order Boussinesq Equation |
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