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


SIGMA 2 (2006), 041, 5 pages      math.AP/0604278      http://dx.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.

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