### Abstract

Original language | English |
---|---|

Pages (from-to) | 1-21 |

Number of pages | 21 |

Journal | Symmetry, Integrability and Geometry: Methods and Applications |

Volume | 9 |

Issue number | 24 |

DOIs | |

Publication status | Published - 2013 |

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*Symmetry, Integrability and Geometry: Methods and Applications*, vol. 9, no. 24, pp. 1-21. https://doi.org/10.3842/SIGMA.2013.024

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

Research output: Contribution to journal › Article

TY - JOUR

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

AU - VASSILIOU, Peter

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.

AB - 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.

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

DO - 10.3842/SIGMA.2013.024

M3 - Article

VL - 9

SP - 1

EP - 21

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

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

SN - 1815-0659

IS - 24

ER -