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

SIGMA 9 (2013), 024, 21 pages      arXiv:1303.4165      https://doi.org/10.3842/SIGMA.2013.024
Contribution to the Special Issue “Symmetries of Differential Equations: Frames, Invariants and Applications”

Cauchy Problem for a Darboux Integrable Wave Map System and Equations of Lie Type

Peter J. Vassiliou
Program in Mathematics and Statistics, University of Canberra, 2601 Australia

Received September 27, 2012, in final form March 12, 2013; Published online March 18, 2013

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.

Key words: wave map; Cauchy problem; Darboux integrable; Lie system; Lie reduction; explicit representation.

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