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Quasi periodic vortex structures in two-dimensional flows in an inviscid incompressible fluid
St. Petersburg Department of Steklov Mathematical Institute .
Halmstad University, School of Information Science, Computer and Electrical Engineering (IDE), Halmstad Embedded and Intelligent Systems Research (EIS), Applied Mathematics and Physics (CAMP).
2005 (English)In: Russian journal of mathematical physics, ISSN 1061-9208, E-ISSN 1555-6638, Vol. 12, no 2, p. 121-156Article in journal (Refereed) Published
Abstract [en]

We consider a two-dimensional steady motion of an inviscid incompressible fluid described by the equation &UDelta; u(x,y) = F(u(x,y)), where u(x,y) is the streamfunction, &UDelta; is the Laplace operator, and F((.)) an arbitrary function measuring the flow vorticity. Apparently, until now, the only way to treat an equation of the above type with nontrivial function F analytically is to use the algebro-geometric method for integrable equations. In particular, we investigate the Cosh-Laplace equation (ChL) &UDelta; u(x,y) = ± 4cosh(u(x,y)) by means of the special technique of finite-gap integration, which allows us to obtain real solutions of the ChL equation by using a Riemann surface with appropriate symmetry. We study the first nontrivial case corresponding to a Riemann surface of genus g = 3. The hydrodynamical interpretation of finite-gap solutions is meaningful, and we try to understand the fluid processes described by these solutions. To this end, we take a Riemann surface with additional symmetry properties. We present four five-parameter families of exact solutions. These solutions are given in terms of Jacobi elliptic functions, which enables us to directly investigate the relevant properties. We also find explicit formulas for the lines of singularity. It is of interest from the point of view of algebraic geometry that the structure of the theta divisor can be described.

Place, publisher, year, edition, pages
New York: John Wiley & Sons, 2005. Vol. 12, no 2, p. 121-156
National Category
Engineering and Technology
Identifiers
URN: urn:nbn:se:hh:diva-3351ISI: 000229603100001Scopus ID: 2-s2.0-20744440314OAI: oai:DiVA.org:hh-3351DiVA, id: diva2:281571
Available from: 2009-12-16 Created: 2009-12-01 Last updated: 2018-03-23Bibliographically approved

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Bordag, Ljudmila A.

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CiteExportLink to record
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Citation style
  • apa
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