Cauchy problem for a Darboux integrable wave map system and equations of Lie type

Peter VASSILIOU

    Research output: Contribution to journalArticle

    3 Citations (Scopus)

    Abstract

    The Cauchy problem for harmonic maps from Minkowski space with its standard flat metric to a certain non-constant curvature Lorentzian 2-metric is studied. The target manifold is distinguished by the fact that the Euler-Lagrange equation for the energy functional is Darboux integrable. The time evolution of the Cauchy data is reduced to an ordinary differential equation of Lie type associated to SL(2) acting on a manifold of dimension 4. This is further reduced to the simplest Lie system: the Riccati equation. Lie reduction permits explicit representation formulas for various initial value problems. Additionally, a concise (hyperbolic) Weierstrass-type representation formula is derived. Finally, a number of open problems are framed.
    Original languageEnglish
    Pages (from-to)1-21
    Number of pages21
    JournalSymmetry, Integrability and Geometry: Methods and Applications
    Volume9
    Issue number24
    DOIs
    Publication statusPublished - 2013

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    Representation Formula
    Cauchy Problem
    Metric
    Euler-Lagrange Equations
    Harmonic Maps
    Minkowski Space
    Riccati Equation
    Energy Functional
    Cauchy
    Initial Value Problem
    Explicit Formula
    Open Problems
    Ordinary differential equation
    Curvature
    Target
    Standards

    Cite this

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    title = "Cauchy problem for a Darboux integrable wave map system and equations of Lie type",
    abstract = "The Cauchy problem for harmonic maps from Minkowski space with its standard flat metric to a certain non-constant curvature Lorentzian 2-metric is studied. The target manifold is distinguished by the fact that the Euler-Lagrange equation for the energy functional is Darboux integrable. The time evolution of the Cauchy data is reduced to an ordinary differential equation of Lie type associated to SL(2) acting on a manifold of dimension 4. This is further reduced to the simplest Lie system: the Riccati equation. Lie reduction permits explicit representation formulas for various initial value problems. Additionally, a concise (hyperbolic) Weierstrass-type representation formula is derived. Finally, a number of open problems are framed.",
    keywords = "Cauchy problem, Darboux integrable, Explicit representation, Lie reduction, Lie system, Wave map, Harmonic maps",
    author = "Peter VASSILIOU",
    year = "2013",
    doi = "10.3842/SIGMA.2013.024",
    language = "English",
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    journal = "Symmetry, Integrability and Geometry: Methods and Applications (SIGMA)",
    issn = "1815-0659",
    publisher = "Department of Applied Research, Institute of Mathematics of National Academy of Science of Ukraine",
    number = "24",

    }

    Cauchy problem for a Darboux integrable wave map system and equations of Lie type. / VASSILIOU, Peter.

    In: Symmetry, Integrability and Geometry: Methods and Applications, Vol. 9, No. 24, 2013, p. 1-21.

    Research output: Contribution to journalArticle

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    T1 - Cauchy problem for a Darboux integrable wave map system and equations of Lie type

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    PY - 2013

    Y1 - 2013

    N2 - The Cauchy problem for harmonic maps from Minkowski space with its standard flat metric to a certain non-constant curvature Lorentzian 2-metric is studied. The target manifold is distinguished by the fact that the Euler-Lagrange equation for the energy functional is Darboux integrable. The time evolution of the Cauchy data is reduced to an ordinary differential equation of Lie type associated to SL(2) acting on a manifold of dimension 4. This is further reduced to the simplest Lie system: the Riccati equation. Lie reduction permits explicit representation formulas for various initial value problems. Additionally, a concise (hyperbolic) Weierstrass-type representation formula is derived. Finally, a number of open problems are framed.

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    KW - Cauchy problem

    KW - Darboux integrable

    KW - Explicit representation

    KW - Lie reduction

    KW - Lie system

    KW - Wave map

    KW - Harmonic maps

    U2 - 10.3842/SIGMA.2013.024

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    JF - Symmetry, Integrability and Geometry: Methods and Applications (SIGMA)

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