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

SIGMA 9 (2013), 024, 21 pages      arXiv:1303.4165
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

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.

pdf (457 kb)   tex (28 kb)


  1. Anderson I.M., Fels M.E., The Cauchy problem for Darboux integrable systems and non-linear d'Alembert formulas, SIGMA 9 (2013), 017, 22 pages, arXiv:1210.2370.
  2. Anderson I.M., Fels M.E., Vassiliou P.J., Superposition formulas for exterior differential systems, Adv. Math. 221 (2009), 1910-1963, arXiv:0708.0679.
  3. Baird P., Wood J.C., Harmonic morphisms between Riemannian manifolds, London Mathematical Society Monographs. New Series, Vol. 29, The Clarendon Press, Oxford University Press, Oxford, 2003.
  4. Bryant R.L., An introduction to Lie groups and symplectic geometry, in Geometry and Quantum Field Theory (Park City, UT, 1991), IAS/Park City Math. Ser., Vol. 1, Amer. Math. Soc., Providence, RI, 1995, 5-181, available at
  5. Cariñena J.F., de Lucas J., Lie systems: theory, generalisations, and applications, Dissertationes Math. 479 (2011), 162 pages, arXiv:1103.4166.
  6. Clelland J.N., Vassiliou P.J., A solvable string on a Lorentzian surface, arXiv:1303.0087.
  7. Dubrov B.M., Komrakov B.P., The constructive equivalence problem in differential geometry, Sb. Math. 191 (2000), 655-681.
  8. Eells J., Sampson J.H., Harmonic mappings of Riemannian manifolds, Amer. J. Math. 86 (1964), 109-160.
  9. Goursat E., Leçons sur l'intégration des équations aux dérivées partielles du seconde ordre á deux variables indépendent. II, Hermann, Paris, 1898.
  10. Gu C.H., On the Cauchy problem for harmonic maps defined on two-dimensional Minkowski space, Comm. Pure Appl. Math. 33 (1980), 727-737.
  11. Guest M.A., Harmonic maps, loop groups, and integrable systems, London Mathematical Society Student Texts, Vol. 38, Cambridge University Press, Cambridge, 1997.
  12. Hélein F., Wood J.C., Harmonic maps, in Handbook of Global Analysis, Editors D. Krupka, D. Saunders, Elsevier, Amsterdam, 2008, 417-491.
  13. Ivey T.A., Landsberg J.M., Cartan for beginners: differential geometry via moving frames and exterior differential systems, Graduate Studies in Mathematics, Vol. 61, Amer. Math. Soc., Providence, RI, 2003.
  14. Kostrigina O.S., Zhiber A.V., Darboux-integrable two-component nonlinear hyperbolic systems of equations, J. Math. Phys. 52 (2011), 033503, 32 pages.
  15. Ream R., Darboux integrability of wave maps into two-dimensional Riemannian manifolds, Masters degree thesis, Utah State University, 2008, available at
  16. Shatah J., Struwe M., Geometric wave equations, Courant Lecture Notes in Mathematics, Vol. 2, New York University Courant Institute of Mathematical Sciences, New York, 1998.
  17. Tao T., Wave maps,
  18. Vassiliou P.J., A constructive generalised Goursat normal form, Differential Geom. Appl. 24 (2006), 332-350, math.DG/0404377.
  19. Vassiliou P.J., Efficient construction of contact coordinates for partial prolongations, Found. Comput. Math. 6 (2006), 269-308, math.DG/0406234.
  20. Vassiliou P.J., Tangential characteristic symmetries and first order hyperbolic systems, Appl. Algebra Engrg. Comm. Comput. 11 (2001), 377-395.
  21. Vassiliou P.J., Vessiot structure for manifolds of (p,q)-hyperbolic type: Darboux integrability and symmetry, Trans. Amer. Math. Soc. 353 (2001), 1705-1739.
  22. Zakrzewski W.J., Low-dimensional sigma models, Adam Hilger Ltd., Bristol, 1989.
  23. Zhiber A.V., Sokolov V.V., Exactly integrable hyperbolic equations of Liouville type, Russ. Math. Surv. 56 (2001), no. 1, 61-101.

Previous article  Next article   Contents of Volume 9 (2013)