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Cauchy Problem for a Darboux Integrable Wave Map System and Equations of Lie Type
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- Symmetry, Integrability and Geometry: Methods and Applications (SIGMA)
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- Symmetry, Integrability and Geometry: Methods and Applications, 2013, том 9
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Cauchy Problem for a Darboux Integrable Wave Map System and Equations of Lie Type
Інший ID:
2010 Mathematics Subject Classification: 53A35; 53A55; 58A15; 58A20; 58A30
DOI: http://dx.doi.org/10.3842/SIGMA.2013.024
DOI: http://dx.doi.org/10.3842/SIGMA.2013.024
Посилання:
Cauchy Problem for a Darboux Integrable Wave Map System and Equations of Lie Type / P.J. Vassiliou // Symmetry, Integrability and Geometry: Methods and Applications. — 2013. — Т. 9. — Бібліогр.: 23 назв. — англ.
Підтримка:
This paper is a contribution to the Special Issue “Symmetries of Dif ferential Equations: Frames, Invariants
and Applications”. The full collection is available at http://www.emis.de/journals/SIGMA/SDE2012.html.
I’m grateful to the three anonymous referees for their close reading of the manuscript and for
making suggestions which considerably improved the paper. I would like to acknowledge, with
my thanks, the early involvement of Jordane Math´e for carefully working together through
the calculations in Section 4 which formed a portion of his internship from the Ecole normale
sup´erieure de Cachan, France. Much of the research for this paper was carried out while I was
a Visiting Fellow at the Mathematical Sciences Institute of the Australian National University,
Canberra. The hospitality of the MSI is gratefully acknowledged. In particular, I thank Mike
Eastwood and the Dif ferential Geometry Group for stimulating discussions.
Дата:
2013
Переглядів:
574
Завантажень:
239
Анотація:
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.
