Dirichlet and Neumann Problems for String Equation, Poncelet Problem and Pell-Abel Equation - Burskii, V.P.; Zhedanov, A.S. (Symmetry, Integrability and Geometry: Methods and Applications, 2006)
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.