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

SIGMA 9 (2013), 017, 22 pages      arXiv:1210.2370
Contribution to the Special Issue “Symmetries of Differential Equations: Frames, Invariants and Applications”

The Cauchy Problem for Darboux Integrable Systems and Non-Linear d'Alembert Formulas

Ian M. Anderson and Mark E. Fels
Utah State University, Logan Utah, USA

Received October 08, 2012, in final form February 20, 2013; Published online February 27, 2013

To every Darboux integrable system there is an associated Lie group G which is a fundamental invariant of the system and which we call the Vessiot group. This article shows that solving the Cauchy problem for a Darboux integrable partial differential equation can be reduced to solving an equation of Lie type for the Vessiot group G. If the Vessiot group G is solvable then the Cauchy problem can be solved by quadratures. This allows us to give explicit integral formulas, similar to the well known d'Alembert's formula for the wave equation, to the initial value problem with generic non-characteristic initial data.

Key words: Cauchy problem; Darboux integrability; exterior differential systems; d'Alembert's formula.

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  1. Anderson I.M., Fels M.E., Bäcklund transformations for Darboux integrable differential systems, arXiv:1108.5443.
  2. Anderson I.M., Fels M.E., Exterior differential systems with symmetry, Acta Appl. Math. 87 (2005), 3-31.
  3. Anderson I.M., Fels M.E., Symmetry reduction of exterior differential systems and Bäcklund transformations for PDE in the plane, Acta Appl. Math. 120 (2012), 29-60, arXiv:1110.5777.
  4. Anderson I.M., Fels M.E., Vassiliou P.J., Superposition formulas for exterior differential systems, Adv. Math. 221 (2009), 1910-1963.
  5. 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.
  6. Bryant R.L., Chern S.S., Gardner R.B., Goldschmidt H.L., Griffiths P.A., Exterior differential systems, Mathematical Sciences Research Institute Publications, Vol. 18, Springer-Verlag, New York, 1991.
  7. Bryant R.L., Griffiths P.A., Characteristic cohomology of differential systems. II. Conservation laws for a class of parabolic equations, Duke Math. J. 78 (1995), 531-676.
  8. Bryant R.L., Griffiths P.A., Hsu L., Hyperbolic exterior differential systems and their conservation laws. I, Selecta Math. (N.S.) 1 (1995), 21-112.
  9. Cariñena J.F., Grabowski J., Marmo G., Superposition rules, Lie theorem, and partial differential equations, Rep. Math. Phys. 60 (2007), 237-258, math-ph/0610013.
  10. Dubrov B.M., Komrakov B.P., The constructive equivalence problem in differential geometry, Sb. Math. 191 (2000), 655-681.
  11. Duzhin S.V., Lychagin V.V., Symmetries of distributions and quadrature of ordinary differential equations, Acta Appl. Math. 24 (1991), 29-57.
  12. Fels M.E., Exterior differential systems with symmetry, in Symmetries and Overdetermined Systems of Partial Differential Equations, IMA Vol. Math. Appl., Vol. 144, Springer, New York, 2008, 351-366.
  13. Gardner R.B., Kamran N., Characteristics and the geometry of hyperbolic equations in the plane, J. Differential Equations 104 (1993), 60-116.
  14. Kobayashi S., Nomizu K., Foundations of differential geometry, Vol. I, Wiley Classics Library, John Wiley & Sons Inc., New York, 1996.
  15. Olver P.J., Applications of Lie groups to differential equations, Graduate Texts in Mathematics, Vol. 107, 2nd ed., Springer-Verlag, New York, 1993.
  16. Vassiliou P.J., Cauchy problem for a Darboux integrable wave map system and equations of Lie type, SIGMA, to appear.

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