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Article: Finiteness of fixed equilibrium configurations of point vortices in the plane with a background flow

TitleFiniteness of fixed equilibrium configurations of point vortices in the plane with a background flow
Authors
Issue Date2014
Citation
Nonlinearity, 2014, v. 27, p. 2445-2463 How to Cite?
AbstractFor a dynamic system consisting of n point vortices in an ideal plane fluid with a steady, incompressible and irrotational background flow, a more physically significant definition of a fixed equilibrium configuration is suggested. Under this new definition, if the complex polynomial w that determines the aforesaid background flow is non-constant, we have found an attainable generic upper bound $\frac{(m+n-1)!}{(m-1)!\,n_1!\cdots n_{i_0}!}$ for the number of fixed equilibrium configurations. Here, m = deg w, i0 is the number of species, and each ni is the number of vortices in a species. We transform the rational function system arising from fixed equilibria into a polynomial system, whose form is good enough to apply the BKK theory (named after Bernshtein (1975 Funct. Anal. Appl. 9 183–5), Khovanskii (1978 Funct. Anal. Appl. 12 38–46) and Kushnirenko (1976 Funct. Anal. Appl. 10 233–5)) to show the finiteness of its number of solutions. Having this finiteness, the required bound follows from Bézout's theorem or the BKK root count by Li and Wang (1996 Math. Comput. 65 1477–84).
Persistent Identifierhttp://hdl.handle.net/10722/202989
ISI Accession Number ID

 

DC FieldValueLanguage
dc.contributor.authorCHEUNG, PLen_US
dc.contributor.authorNg, TWen_US
dc.date.accessioned2014-09-19T11:06:55Z-
dc.date.available2014-09-19T11:06:55Z-
dc.date.issued2014en_US
dc.identifier.citationNonlinearity, 2014, v. 27, p. 2445-2463en_US
dc.identifier.urihttp://hdl.handle.net/10722/202989-
dc.description.abstractFor a dynamic system consisting of n point vortices in an ideal plane fluid with a steady, incompressible and irrotational background flow, a more physically significant definition of a fixed equilibrium configuration is suggested. Under this new definition, if the complex polynomial w that determines the aforesaid background flow is non-constant, we have found an attainable generic upper bound $\frac{(m+n-1)!}{(m-1)!\,n_1!\cdots n_{i_0}!}$ for the number of fixed equilibrium configurations. Here, m = deg w, i0 is the number of species, and each ni is the number of vortices in a species. We transform the rational function system arising from fixed equilibria into a polynomial system, whose form is good enough to apply the BKK theory (named after Bernshtein (1975 Funct. Anal. Appl. 9 183–5), Khovanskii (1978 Funct. Anal. Appl. 12 38–46) and Kushnirenko (1976 Funct. Anal. Appl. 10 233–5)) to show the finiteness of its number of solutions. Having this finiteness, the required bound follows from Bézout's theorem or the BKK root count by Li and Wang (1996 Math. Comput. 65 1477–84).en_US
dc.languageengen_US
dc.relation.ispartofNonlinearityen_US
dc.rightsCreative Commons: Attribution 3.0 Hong Kong License-
dc.titleFiniteness of fixed equilibrium configurations of point vortices in the plane with a background flowen_US
dc.typeArticleen_US
dc.identifier.emailNg, TW: ngtw@hku.hken_US
dc.identifier.authorityNg, TW=rp00768en_US
dc.description.naturepostprint-
dc.identifier.doi10.1088/0951-7715/27/10/2445en_US
dc.identifier.hkuros239026en_US
dc.identifier.volume27en_US
dc.identifier.spage2445en_US
dc.identifier.epage2463en_US
dc.identifier.isiWOS:000342751000002-

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