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

SIGMA 2 (2006), 041, 5 pages      math.AP/0604278      https://doi.org/10.3842/SIGMA.2006.041

Dirichlet and Neumann Problems for String Equation, Poncelet Problem and Pell-Abel Equation

Vladimir P. Burskii a and Alexei S. Zhedanov b
a) Institute of Applied Mathematics and Mechanics NASU, Donetsk, 83114 Ukraine
b) Donetsk Institute for Physics and Technology NASU, Donetsk, 83114 Ukraine

Received November 23, 2005, in final form March 20, 2006; Published online April 12, 2006

Abstract
We consider conditions for uniqueness of the solution of the Dirichlet or the Neumann problem for 2-dimensional wave equation inside of bi-quadratic algebraic curve. We show that the solution is non-trivial if and only if corresponding Poncelet problem for two conics associated with the curve has periodic trajectory and if and only if corresponding Pell-Abel equation has a solution.

Key words: Dirichlet problem; Neumann problem; string equation Poncelet problem; Pell-Abel equation.

pdf (155 kb)   ps (126 kb)   tex (9 kb)

References

  1. Baxter R., Exactly solvable models in statistical mechanics, London, Academic Press, 1982.
  2. Berger M., Géométrie, CEDIC, Paris, 1978.
  3. Burskii V.P., Investigation methods of boundary value problems for general differential equations, Kyiv, Naukova Dumka, 2002 (in Russian).
  4. Burskii V.P., Zhedanov A.S., The Dirichlet and the Poncelet problems, RIAM Symposium No.16ME-S1 "Physics and Mathematical Structures of Nonlinear Waves" (November 15-17, 2004, Kyushu University, Kasuga, Fukuoka, Japan), 2004, 22-26, see here.
  5. Francoise J.P., Ragnisco O., An iterative process on quartics and integrable symplectic maps, in Symmetries and Integrability of Difference equations (1996, Canterbury), Editors P.A. Clarkson and F.W. Nijhoff, London Math. Soc. Lecture Note Ser., Vol. 255, Cambridge, Cambridge Univ. Press, 1999, 56-63.
  6. Granovskii Ya.I., Zhedanov A.S., Integrability of the classical XY chain, Pis'ma Zh. Eksper. Teoret. Fiz., 1986, V.44, 237-239 (English transl.: JETP Lett., 1986, V.44, N 5, 304-307).
  7. Griffiths P., Harris J., On a Cayley's explicit solution to Poncelet's porism, Enseign. Math. 2, 1978, V.24, 31-40.
  8. Iatrou A., Roberts J.A.G., Integrable mappings of the plane preserving biquadratic invariant curves II, Nonlinearity, 2002, V.15, 459-489.
  9. John F., The Dirichlet problem for a hyperbolic equation, Amer. J. Math., 1941, V.63, 141-154.
  10. Malyshev V.A., Abel equation, Algebra i Analiz, 2001, V.13, N 6, 1-55 (English transl.: St. Petersburg Math. J., 2001, V.13, N 6, 893-938).
  11. Malyshev V.A., Periods of quadratic irrationalities, and torsion of elliptic curves, Algebra i Analiz, 2003, V.15, N 4, 177-203 (English transl.: St. Petersburg Math. J., 2004, V.15, N 4, 587-602).
  12. Sodin M.L., Yuditskii P.M., Functions least deviating from zero on closed subsets of the real axis, Algebra i Analiz, 1992, V.4, N 2, 1-61 (English transl.: St. Petersburg Math. J., 1993, V.4, N 2, 201-249).
  13. Veselov A.P., Integrable systems with discrete time and difference operators, Funktsional. Anal. i Prilozhen., 1988, V.22, 1-13 (English transl.: Funct. Anal. Appl., 1988, V.22, N 2, 83-93).
  14. Whittacker E.T., Watson G.N., A course of modern analysis, Cambridge, 1927.

Previous article   Next article   Contents of Volume 2 (2006)