Three-dimensional gravity–capillary waves on water — Small surface tension case
This paper considers three-dimensional gravity–capillary waves on water of finite-depth, which are uniformly translating in a horizontal propagating direction and periodic in a transverse direction. The exact Euler equations are formulated as a spatial dynamic system in which the variable used for t...
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description | This paper considers three-dimensional gravity–capillary waves on water of finite-depth, which are uniformly translating in a horizontal propagating direction and periodic in a transverse direction. The exact Euler equations are formulated as a spatial dynamic system in which the variable used for the propagating direction is a time-like variable. The existence of the solutions of the system is determined by two non-dimensional constants, the Bond number
b
and the Froude number
F
, which in turn give the number of eigenvalues on the imaginary axis of the complex plane for the corresponding linearized operator around a uniform flow. Assume that
λ
=
F
−
2
,
C
1
is the curve in the
(
b
,
λ
)
-plane on which the first two eigenvalues for three-dimensional waves collide at the imaginary axis, and the intersection point of
C
1
with
{
λ
=
1
}
is
b
1
>
0
. In this paper, the case for
0
<
b
<
b
1
and
(
b
,
λ
)
near
C
1
is considered. A center-manifold reduction technique and a normal form analysis are applied to show that the dynamical system can be reduced to a system of ordinary differential equations. Using the existence of a homoclinic orbit connecting to a two-dimensional periodic (called generalized solitary-wave, thereafter) solution for the reduced system, it is shown that such a generalized solitary-wave solution persists for the original system by applying a perturbation method and adjusting some appropriate constants. |
doi_str_mv | 10.1016/j.physd.2009.05.012 |
format | Article |
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b
and the Froude number
F
, which in turn give the number of eigenvalues on the imaginary axis of the complex plane for the corresponding linearized operator around a uniform flow. Assume that
λ
=
F
−
2
,
C
1
is the curve in the
(
b
,
λ
)
-plane on which the first two eigenvalues for three-dimensional waves collide at the imaginary axis, and the intersection point of
C
1
with
{
λ
=
1
}
is
b
1
>
0
. In this paper, the case for
0
<
b
<
b
1
and
(
b
,
λ
)
near
C
1
is considered. A center-manifold reduction technique and a normal form analysis are applied to show that the dynamical system can be reduced to a system of ordinary differential equations. Using the existence of a homoclinic orbit connecting to a two-dimensional periodic (called generalized solitary-wave, thereafter) solution for the reduced system, it is shown that such a generalized solitary-wave solution persists for the original system by applying a perturbation method and adjusting some appropriate constants.</description><identifier>ISSN: 0167-2789</identifier><identifier>EISSN: 1872-8022</identifier><identifier>DOI: 10.1016/j.physd.2009.05.012</identifier><identifier>CODEN: PDNPDT</identifier><language>eng</language><publisher>Amsterdam: Elsevier B.V</publisher><subject>Exact sciences and technology ; Homoclinic orbits ; Normal form ; Periodic orbits ; Physics ; Three-dimensional solitary waves</subject><ispartof>Physica. D, 2009-08, Vol.238 (17), p.1735-1751</ispartof><rights>2009</rights><rights>2009 INIST-CNRS</rights><lds50>peer_reviewed</lds50><woscitedreferencessubscribed>false</woscitedreferencessubscribed><citedby>FETCH-LOGICAL-c364t-101a72d4f19aec69a3111ee765fe1b37867079bea3eb43094b61c8fbbd9cae323</citedby><cites>FETCH-LOGICAL-c364t-101a72d4f19aec69a3111ee765fe1b37867079bea3eb43094b61c8fbbd9cae323</cites></display><links><openurl>$$Topenurl_article</openurl><openurlfulltext>$$Topenurlfull_article</openurlfulltext><thumbnail>$$Tsyndetics_thumb_exl</thumbnail><linktohtml>$$Uhttps://dx.doi.org/10.1016/j.physd.2009.05.012$$EHTML$$P50$$Gelsevier$$H</linktohtml><link.rule.ids>314,780,784,3550,27924,27925,45995</link.rule.ids><backlink>$$Uhttp://pascal-francis.inist.fr/vibad/index.php?action=getRecordDetail&idt=21824988$$DView record in Pascal Francis$$Hfree_for_read</backlink></links><search><creatorcontrib>Deng, Shengfu</creatorcontrib><creatorcontrib>Sun, Shu-Ming</creatorcontrib><title>Three-dimensional gravity–capillary waves on water — Small surface tension case</title><title>Physica. D</title><description>This paper considers three-dimensional gravity–capillary waves on water of finite-depth, which are uniformly translating in a horizontal propagating direction and periodic in a transverse direction. The exact Euler equations are formulated as a spatial dynamic system in which the variable used for the propagating direction is a time-like variable. The existence of the solutions of the system is determined by two non-dimensional constants, the Bond number
b
and the Froude number
F
, which in turn give the number of eigenvalues on the imaginary axis of the complex plane for the corresponding linearized operator around a uniform flow. Assume that
λ
=
F
−
2
,
C
1
is the curve in the
(
b
,
λ
)
-plane on which the first two eigenvalues for three-dimensional waves collide at the imaginary axis, and the intersection point of
C
1
with
{
λ
=
1
}
is
b
1
>
0
. In this paper, the case for
0
<
b
<
b
1
and
(
b
,
λ
)
near
C
1
is considered. A center-manifold reduction technique and a normal form analysis are applied to show that the dynamical system can be reduced to a system of ordinary differential equations. Using the existence of a homoclinic orbit connecting to a two-dimensional periodic (called generalized solitary-wave, thereafter) solution for the reduced system, it is shown that such a generalized solitary-wave solution persists for the original system by applying a perturbation method and adjusting some appropriate constants.</description><subject>Exact sciences and technology</subject><subject>Homoclinic orbits</subject><subject>Normal form</subject><subject>Periodic orbits</subject><subject>Physics</subject><subject>Three-dimensional solitary waves</subject><issn>0167-2789</issn><issn>1872-8022</issn><fulltext>true</fulltext><rsrctype>article</rsrctype><creationdate>2009</creationdate><recordtype>article</recordtype><recordid>eNp9kLtOw0AQRVcIJELgC2jcQGezD8e7LihQxEuKRJFQr8brMdnIscOuE5Qu_wBfmC9h8xAl1Uxx7p25l5BrRhNGWXY3SxbTtS8TTmme0EFCGT8hPaYkjxXl_JT0AiVjLlV-Ti68n1FKmRSyR8aTqUOMSzvHxtu2gTr6cLCy3Xq7-TawsHUNbh19wQp91DZh6dBF281PNJ5DXUd-6SowGHUHeWTA4yU5q6D2eHWcffL-9DgZvsSjt-fX4cMoNiJLuzh8DpKXacVyQJPlIBhjiDIbVMgKIVUmqcwLBIFFKmieFhkzqiqKMjeAgos-uT34Llz7uUTf6bn1BsPHDbZLr0WqBFN7UBxA41rvHVZ64ew85NKM6l2Beqb3BepdgZoOdCgwqG6O9uAN1JWDxlj_J-VM8TRXKnD3Bw5D1pVFp72x2BgsrUPT6bK1_975BWheiow</recordid><startdate>20090815</startdate><enddate>20090815</enddate><creator>Deng, Shengfu</creator><creator>Sun, Shu-Ming</creator><general>Elsevier B.V</general><general>Elsevier</general><scope>IQODW</scope><scope>AAYXX</scope><scope>CITATION</scope><scope>7U5</scope><scope>8FD</scope><scope>L7M</scope></search><sort><creationdate>20090815</creationdate><title>Three-dimensional gravity–capillary waves on water — Small surface tension case</title><author>Deng, Shengfu ; Sun, Shu-Ming</author></sort><facets><frbrtype>5</frbrtype><frbrgroupid>cdi_FETCH-LOGICAL-c364t-101a72d4f19aec69a3111ee765fe1b37867079bea3eb43094b61c8fbbd9cae323</frbrgroupid><rsrctype>articles</rsrctype><prefilter>articles</prefilter><language>eng</language><creationdate>2009</creationdate><topic>Exact sciences and technology</topic><topic>Homoclinic orbits</topic><topic>Normal form</topic><topic>Periodic orbits</topic><topic>Physics</topic><topic>Three-dimensional solitary waves</topic><toplevel>peer_reviewed</toplevel><toplevel>online_resources</toplevel><creatorcontrib>Deng, Shengfu</creatorcontrib><creatorcontrib>Sun, Shu-Ming</creatorcontrib><collection>Pascal-Francis</collection><collection>CrossRef</collection><collection>Solid State and Superconductivity Abstracts</collection><collection>Technology Research Database</collection><collection>Advanced Technologies Database with Aerospace</collection><jtitle>Physica. D</jtitle></facets><delivery><delcategory>Remote Search Resource</delcategory><fulltext>fulltext</fulltext></delivery><addata><au>Deng, Shengfu</au><au>Sun, Shu-Ming</au><format>journal</format><genre>article</genre><ristype>JOUR</ristype><atitle>Three-dimensional gravity–capillary waves on water — Small surface tension case</atitle><jtitle>Physica. D</jtitle><date>2009-08-15</date><risdate>2009</risdate><volume>238</volume><issue>17</issue><spage>1735</spage><epage>1751</epage><pages>1735-1751</pages><issn>0167-2789</issn><eissn>1872-8022</eissn><coden>PDNPDT</coden><abstract>This paper considers three-dimensional gravity–capillary waves on water of finite-depth, which are uniformly translating in a horizontal propagating direction and periodic in a transverse direction. The exact Euler equations are formulated as a spatial dynamic system in which the variable used for the propagating direction is a time-like variable. The existence of the solutions of the system is determined by two non-dimensional constants, the Bond number
b
and the Froude number
F
, which in turn give the number of eigenvalues on the imaginary axis of the complex plane for the corresponding linearized operator around a uniform flow. Assume that
λ
=
F
−
2
,
C
1
is the curve in the
(
b
,
λ
)
-plane on which the first two eigenvalues for three-dimensional waves collide at the imaginary axis, and the intersection point of
C
1
with
{
λ
=
1
}
is
b
1
>
0
. In this paper, the case for
0
<
b
<
b
1
and
(
b
,
λ
)
near
C
1
is considered. A center-manifold reduction technique and a normal form analysis are applied to show that the dynamical system can be reduced to a system of ordinary differential equations. Using the existence of a homoclinic orbit connecting to a two-dimensional periodic (called generalized solitary-wave, thereafter) solution for the reduced system, it is shown that such a generalized solitary-wave solution persists for the original system by applying a perturbation method and adjusting some appropriate constants.</abstract><cop>Amsterdam</cop><pub>Elsevier B.V</pub><doi>10.1016/j.physd.2009.05.012</doi><tpages>17</tpages></addata></record> |
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language | eng |
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source | Elsevier ScienceDirect Journals Complete |
subjects | Exact sciences and technology Homoclinic orbits Normal form Periodic orbits Physics Three-dimensional solitary waves |
title | Three-dimensional gravity–capillary waves on water — Small surface tension case |
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