Vladimir P. Burskii, Alexei S. Zhedanov, “Dirichlet and Neumann Problems for String Equation, Poncelet Problem and Pell–Abel Equation”, SIGMA, 2006, Volume 2,041, 5 pp.

This article is cited in 4 papers

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

Vladimir P. Burskii^a, 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

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.

Keywords: Dirichlet problem; Neumann problem; string equation; Poncelet problem; Pell–Abel equation.

MSC: 35L20; 14H70; 13B25

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

Language: English

DOI: 10.3842/SIGMA.2006.041