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

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.

We will examine the Dirichlet problem for the string equation in a bounded semialgebraic domain, the boundary of which is given by some bi-quadratic algebraic curve given in the Euler-Baxter form [1,13]: F (x, y) = x 2 y 2 + 1 + a x 2 + y 2 + 2bxy = 0 with real a and b that are subordinated to conditions of boundedness a > 0 and nonvanishing (|b| − a) 2 > 1. The Euler-Baxter curve (3) is a special case of generic bi-quadratic curve 2 i,k=0 a ik x i y k = 0 (4) with arbitrary 9 coefficients a ik . In the general situation this curve has a genus g = 1 and hence is birationally equivalent to an elliptic curve [7]. As far as we know, the John algorithm for such curve was first considered in [5], where it was shown that the corresponding discrete dynamical system is completely integrable and some solutions in terms of elliptic functions were presented in special cases.
In this paper we will concentrate mostly on analysis of the Euler-Baxter curve (3), because the generic curve (4) can be transformed to the form (3) by elementary linear rational transformations [8] x → Recall that the John algorithm is defined as follows [9].
Let Ω be an arbitrary bounded domain, which is convex with respect to characteristic directions, i.e. it has the boundary C intersected in at most two points by each line that is parallel to x-or y-axes. We start from an arbitrary point M 1 on C and consider a vertical line passing through M 1 . Obviously, there are exactly two points of intersection with the curve C: M 1 and some M 2 . We denote I 1 an involution which transform M 1 into M 2 . Then, starting from M 2 , we consider a horizontal line passing through M 2 . Let M 3 be the second point of intersection with the curve C. Let I 2 be corresponding involution: I 2 M 2 = M 3 . We then repeat this process, applying step-by-step involutions I 1 and I 2 . Denote T = I 2 I 1 , T −1 = I 1 I 2 . This transformation T : C → C produces a discrete dynamical system on C, i.e. an action of group Z and each point M ∈ C generates an orbit {T n M | n ∈ Z}. This orbit can be either finite or infinite. The point M corresponding to a finite orbit is called a periodic point and minimal n, for which T n M = M , is called a period of the point M . F. John have proved several useful assertions, among which we point out the following one.
Sufficient condition of uniqueness. The homogeneous Dirichlet problem has no solutions apart form the trivial one in the space C 2 (Ω) if the set of periodic points of T on C is finite or denumerable, in particular, if there are no periodic points on C.
Recall now the Poncelet algorithm [2]. Let A and B be two arbitrary conics (say, two ellipses, this cases is general and can be achieved by an appropriate projective transformation). We start from an arbitrary point M 1 on the conic B and pass a tangent to the conic A. This tangent intersects the conic B in another point M 2 . Then we pass another tangent from the point M 2 to the conic A. We obtain then the point M 3 on the conic B. This process can be repeated generating a set of the points M 1 , M 2 , . . . , M n , . . . on the conic B. We denote also L 1 , L 2 , . . . corresponding tangential points on the conic A. The famous Poncelet theorem [2] states that if this algorithm is periodic for M 1 (i.e. Our crucial observation is the following. It appears that the John algorithm for the biquadratic curve C is equivalent to the Poncelet algorithm for some two conics in a two-dimensional affine space. Consider a transformation V : (x, y) → (u, v) given by the formulae u = x + y; v = xy. This transformation converts the bisectrix x = y into the parabola D = {(u, v) | v = u 2 /4}, moreover the left halfplane x < y is transformed onto the exterior D − of the parabola D and the mapping is a diffeomorphism, the right halfplane x > y is transformed onto the same exterior D − of the parabola D, the points (x, y) and (y, x) have the same image. The set of vertical lines 0 , having the parabola D as their envelope. The set of horizontal lines y = y 0 is transformed also into the same set E of lines (the lines x = c and y = c have the same image). The curve (3) is transformed into the quadric B: Note that in our previous paper [4] we proposed a slightly different way to establish a connection between the Poncelet problem and the boundary value problem (1), (2).
This result allows one to establish some important properties of the John algorithm for our curve F (u, v) = 0. For example, the Poncelet theorem says then that the periodicity property of the John algorithm depends only on equation of the bi-quadratic curve (3) and does not depend on the choice of initial point on this curve.
This means that only one of two possibilities occurs: (i) either all points of the curve C are non-periodic. It follows from the Poncelet problem that in this case the set T n M is dense on C, it is said to be the transitive case; (ii) or all points of the bi-quadratic curve (3) have the same finite period N . Moreover, we can establish an explicit criterion for periodicity condition for the John algorithm (see below) and to demonstrate a relation with the famous algebraic Pell-Abel equation. This relation comes from the well-known Cayley criterion for the Poncelet problem. Recall that the Cayley criterion can be formulated as follows [2]. Let f (λ) = det(A − λB) be a characteristic determinant for the one-parameter pencil of conics A and B presented in the projective form. In more details, assume that the conic A has an affine equation φ A (x, y) = 0. We then pass to the projective co-ordinates x = ξ 1 /ξ 0 , y = ξ 2 /ξ 0 and present the equation of the conic A in the form f (λ) = c 0 + c 1 λ + · · · + c n λ n + · · · and compute the Hankel-type determinants from these Taylor coefficients: and Then the Cayley criterion [2,7] is: the trajectory of the Poncelet problem is periodic with the period N if and only if H  [7]. Moreover, we have the following observation: the Cayley condition (6) coincides with a solvability criterion of the Pell-Abel equation with degf = 4, f (0) = 0 (for details concerning solvability of the Pell-Abel equation and its relations with other problems of mathematics see, e.g. papers by V.A. Malyshev [10,11]) if one takes f (x) = x 4f (x −1 ). Recall that in the equation (8) the function f is known and one should find unknown polynomials A and B. The main problem here is to find conditions on the coefficient f under which the equation has a solution A, B. We have the following proposition (a detailed analysis of this proposition will be published elsewhere): The Poncelet problem is periodic with an even period iff corresponding the Pell-Abel equation is solvable.
Note that the case of odd period 2N + 1 for the Poncelet problem corresponds to the case of the same period 2N + 1 for the John algorithm, whereas the case of even period 2N for the Poncelet problem corresponds to the case of period N for the John algorithm.
Thus, in order to decide whether or not the John algorithm for the given bi-quadratic curve (3) is periodic we first pass from the John algorithm to the corresponding Poncelet problem and then apply the Cayley criterion.
Moreover, we have given the explicit solution of the John algorithm for the bi-quadratic curve (4) (see [4]): x n = φ(q(n − n 1 )), y n = ψ(q(n − n 2 )), where φ(z), ψ(z) are two different elliptic functions of the second order with the same periods; n 1 , n 2 are some parameters depending on initial conditions. The parameter q and the periods 2ω 1 , 2ω 2 of the elliptic functions φ(z), ψ(z) do not depend on initial conditions. These considerations allow one to write down the general solution of the John algorithm in the form Periodicity condition for the John algorithm is where N , m 1 , m 2 are integers. For the special case of the Euler-Baxter biquadratic curve (3) we have a much simpler solution x n = √ k sn(q(n − n 0 ), k), y n = √ k sn(q(n − n 0 + 1/2), k), where n 0 is an arbitrary parameter and parameters q, k are easily determined from the parameters a, b of the Euler-Baxter curve [1,8]. Finally, we obtained new results concerning solutions for the boundary value problems for the string equation in case of periodicity of the John algorithm. For the Dirichlet problem we have the following result: Let C be a nondegenerate bi-quadratic curve (3). The homogeneous Dirichlet problem (1), (2) has a nontrivial solution Φ ∈ C 2 (Ω) iff the John algorithm is periodic, i.e. the condition (10) is fulfilled. In this case there is an infinite set of linearly independent smooth solutions.
Besides, note the Dirichlet problem (2) has some nontrivial solution u in Sobolev spaces iff the Neumann problem u ′ ν * | C = 0 with the conormal ν * for the equation (1) has a nonconstant solution u [3].
Thus, the Poncelet problem and Pell-Abel equation are closely connected to the Neumann problem for the string equation as well. And the condition (10) is a criterion of existence of nontrivial solution for each of these problems.