On Dirichlet problem for string equation, Poncelet problem, Pell-Abel equation, and some other related problems

Abstract

UDC 517.95In a plane domain bounded by a biquadratic curve, we consider the problem of the uniqueness of a solution of the Dirichlet problem for the string equation. We show that this problem is equivalent to the classical Poncelet problem in projective geometry for two appropriate ellipses and also to the problem of the solvability of the Pell -Abel algebraic equation; some other related problems are also considered.In the present paper, we describe the relationships between properties of boundary-value problems for the string equation in a domain bounded by a biquadratic curve and some classical problems in algebra, geometry, and analysis that have recently been established by the authors. Boundary-Value Problems1.1. Historical Information. Investigations of ill-posed boundary-value problems date back to Hadamard [1], who first noted that the homogeneous Dirichlet problem for the string equation in a certain rectangle admits a nontrivial solution. The investigations carried out by Huber [2] and then by Bourgin and Duffin [3] resulted in the establishment of a condition for the violation of the uniqueness of a solution of the homogenous Dirichlet problem for the equation u tt -u xx = 0 in the rectangle { 0 ≤ t ≤ T; 0 ≤ x ≤ X }. It turned out that the uniqueness of a solution of this problem in classical spaces is violated if and only if the ratio T / X is rational. In the papers indicated, the method of Fourier expansion of a solution was used. In the works of Ptashnik and his disciples, this method was used for the investigation of properties of various boundary-value problems for partial differential equations in a rectangle and a parallelepiped [4]. In [5], John studied the problem of the violation of the uniqueness of a solution of the homogeneous Dirichlet problem for the string equation in a general bounded plane domain convex with respect to both families of characteristics in connection with a certain mapping of the boundary of the domain into itself (a so-called characteristic billiard, see below) used earlier in the mentioned works of Hadamard and Huber. The investigation of this relationship was continued by Aleksandryan and his disciples in the case of the Sobolev problem of oscillations of the surface of a liquid filling a cavity in a flying body [6 -8]. In [9], Arnol'd showed the relationship between this Dirichlet problem in an ellipse and the problem of small denominators in the context of the influence of the rate of approximation of a certain number associated with the problem by rational numbers on the smoothness of a solution. Some questions related to these problems were studied by Berezanskii [10], Zelenyak [11], Fokin [12], etc.Let us describe a characteristic billiard. Consider the Dirichlet problem for the string equation

Poncelet problem for rational conics

“…In the paper [2], a relationship was established between the Poncelet problem and the Abel equation. The point is that, for polygons with an even number of vertices, the Cayley solvability criterion for the Poncelet problem is identical to the solvability criterion for the Abel equation in terms of the Hankel determinants [3].…”

On Dirichlet, Poncelet and Abel problems

“…Note that some results of the present work were already published in the form of brief and incomplete fragments in papers [18], [19] and [20].…”

“…In a similar way, all other cases can also be considered (it was done in works [18], [11], [55]). Thus, we will formulate the final result as follows: There are also three exceptional cases connected with elliptic functions with certain restrictions imposed upon their parameters:…”

“…Thus, we obtain the main result Proposition 25. The Poncelet problem in a generic setting is periodic if and only if the John mapping on the corresponding biquadratic curve (4.5) is periodic, and then their periods are coincide.Explicit answers for the John mapping were given above, in propositions15,18,19. We obtainProposition 26.…”

The Poncelet problem and the discrete-time pendulum

“…Recently, it has been shown [7] that the Poncelet theorem plays an important role in the analysis of the Dirichlet problem for the string equation.…”