We prove that in quadratic perturbations of generic quadratic Hamiltonian vector fields with three saddle points and one centre there can appear at most two limit cycles. This bound is exact.
We develop a systematic way for constructing bispectral algebras of commuting ordinary differential operators of any rank N . It combines and unifies the ideas of Duistermaat-Grünbaum and Wilson. Our construction is completely algorithmic and enables us to obtain all previously known classes or individual examples of bispectral operators. The method also provides new broad families of bispectral algebras which may help to penetrate deeper into the problem.
We present methods for obtaining new solutions to the bispectral problem. We achieve this by giving its abstract algebraic version suitable for generalizations. All methods are illustrated by new classes of bispectral operators.q-alg/9605011
IntroductionThe bispectral problem of J. J. Duistermaat and F. A. Grünbaum [9] consists of finding all bispectral ordinary differential operators, i.e. operators L(x, ∂ x ) having a family of eigenfunctions ψ(x, z), which are also eigenfunctions for another differential operator Λ(z, ∂ z ) in the spectral parameter:In [3] we constructed large families of solutions to this problem, generalizing all previously known results (cf. [9,19,16,17,13], etc.). We obtained them as special Darboux transformations (called "polynomial") from the most obvious solutionsthe (generalized) Bessel and Airy ones. We recall that the Bessel (respectivelyAiry) operators have the formFor fixed ψ, all operators L(x, ∂ x ) for which ψ is an eigenfunction form a commutative algebra A ψ , called the spectral algebra. Then SpecA ψ is an algebraic curve [6] -the spectral curve. The dimension of the space of eigenfunctions ψ (= g.c.d. ordL, L ∈ A ψ ) is called a rank of A ψ . Following G. Wilson [17] we call the spectral algebra A ψ bispectral iff there exists an operator Λ(z, ∂ z ) satisfying (0.2).The purpose of this paper is to present general methods for constructing bispectral operators (respectively -algebras). Some of our statements are only abstract *
An explicit upper bound Z(3, n) 5n+15 is derived for the number of the zeros of the integral h → I (h) = H =h g(x, y) dx −f (x, y) dy of degree n polynomials f, g, on the open interval for which the cubic curve {H = h} contains an oval. The proof exploits the properties of the Picard-Fuchs system satisfied by the four basic integrals H
Let KdV stand for the Nth Gelfand-Dickey reduction of the KP hierarchy. The purpose of this paper is to show that any KdV solution leads effectively to a solution of the q-approximation of KdV. Two different q-KdV approximations were proposed, first one by E. Frenkel [7] and a variation by Khesin, Lyubashenko and Roger [10]. We show there is a dictionary between the solutions of q-KP and the 1-Toda lattice equations, obeying some special requirement; this is based on an algebra isomorphism between difference operators and D-operators, where Df (x) = f (qx). Therefore every notion about the 1-Toda lattice can be transcribed into q-language.Consider the q-difference operators D and D q , defined by Df (y) = f (qy) and D q f (y) := f (qy) − f (y) (q − 1)y , and the q-pseudo-differential operators
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