Terms Of Theta Functions Research Articles

PERIODIC PEAKONS AND CALOGERO–FRANÇOISE FLOWS

It has long been known that a number of periodic completely integrable systems are associated to hyperelliptic curves, for which the Abel map linearizes the flow (at least in part). We show that this is true for a relatively recent such system: the periodic discrete reduction of the shallow water equation derived by Camassa and Holm. The associated spectral problem has the same form and evolves in the same way as the spectral problem for a family of finite-dimensional non-periodic Hamiltonian flows introduced by Calogero and Françoise. We adapt the Weyl function method used earlier by us to solve the peakon problem to give an explicit solution to both the periodic discrete Camassa–Holm system and the (non-periodic) Calogero–Françoise system in terms of theta functions. AMS 2000 Mathematics subject classification: Primary 35Q51; 37J35; 35Q53

Analytic representation of finite quantum systems

A transform between functions in R and functions in Zd is used to define the analogue of number and coherent states in the context of finite d-dimensional quantum systems. The coherent states are used to define an analytic representation in terms of theta functions. All states are represented by entire functions with growth of order 2, which have exactly d zeros in each cell. The analytic function of a state is constructed from its zeros. Results about the completeness of finite sets of coherent states within a cell are derived.

Inverse functions of Grötzsch’s and Teichmüller’s modulus functions

Let $\chi$ be the inverse of the Grötzsch modulus function and let $\sigma _{n}$ be the $n$-th iteration of the function $\sigma (r) = 2\sqrt{r}/(1 + r)$, $r > 0$. For a real constant $\beta \neq 0$ with $\beta >-2$, the difference $\chi (x)^{\beta}-\sigma _{n}(4e^{-2^{n}x})^{\beta}$ is estimated. In the particular case where $\beta =-2$ one has an approximation of the inverse $S$ of the Teichmüller modulus function, which is applied to improving the known upper and lower estimates concerning the error term of $\lambda (K) = \chi (\pi K/2)^{-2} -1$ from $16^{-1}e^{\pi K}-2^{-1}$ for the variable $K \geqslant 1$. Expressions of $\chi$ and $S$ in terms of theta functions are studied. Lipschitz continuity of $f$ or log $f$ for $f = \chi , S$, as well as other functions are proved.

INVARIANTS FOR SOME REAL HYPERBOLIC GROUPS

For several discontinuous groups acting on the hyperbolic 3-space, which are commensurable with the group arising from the Whitebread link complement, we construct systems of invariant functions, in terms of theta functions, which give explicit projective embeddings of the quotient spaces.

A generalization of the Chebyshev polynomials

In this paper we study polynomials that are orthogonal with respect to a weight function which is zero on a set of positive measure. These were initially introduced by Akhiezer as a generalization of the Chebyshev polynomials where the interval of orthogonality is [-1,α]∪[β,1]. Here, this concept is extended and the interval is the union of g + 1 disjoint intervals, [-1,α1]∪j = 1g-1[βj,αj + 1]∪[βg,1], denoted by E. Starting from a suitably chosen weight function p, and the three-term recurrence relation satisfied by the polynomials, a hyperelliptic Riemann surface is defined, from which we construct representations for both the polynomials of the first (Pn) and second kind (Qn), respectively, in terms of the Riemann theta function of the surface. Explicit expressions for the recurrence coefficients an and bn are found in terms of theta functions. The second-order ordinary differential equation, where Pn and Qn/w (where w is the Stieltjes transform of the weight) are linearly independent solutions, is found. The simpler case, where g = 1, is extensively dealt with and the reduction to the Chebyshev polynomials in the limiting situation, α→β, where the two intervals merge into one, is demonstrated. We also show that p(x)kn(x,x)/n for xE, where kn(x,x) is the reproducing kernel at coincidence, tends to the equilibrium density of the set E, as n→∞.

On Gunning’s Prime Form in Genus 2

AbstractUsing a classical generalization of Jacobi’s derivative formula, we give an explicit expression for Gunning’s prime form in genus 2 in terms of theta functions and their derivatives.

The Elliptic Gaudin System with Spin

We study the algebraic–geometric structure of the elliptic Gaudin two-puncture model previously obtained. We identify this system with the system of pole dynamics of finite-gap solutions of the matrix Davey–Stewartson equation. We also obtain the action–angle variables and construct explicit solutions of this system in terms of theta functions. We discuss the geometry of degenerations of this system.

Isoenergy Spectral Problem for Multidimensional Difference Operators

In this paper, the direct and inverse isoenergy spectral problems are solved for a class of multidimensional periodic difference operators. It is proved that the inverse spectral problem is solvable in terms of theta functions of curves added to the spectral variety under compactification, and multidimensional analogs of the Veselov-Novikov relations are found.

Hele-Shaw flows with free boundaries in a corner or around a wedge Part II: Air at the vertex

Consider a Hele-Shaw cell with the fluid (liquid) confined to an angular region by a solid boundary in the form of two half-lines meeting at an angle απ; if 0 < α < 1 we have flow in a corner, while if 1 < α [les ] 2 we have flow around a wedge. We suppose contact between the fluid and each of the lines forming the solid boundary to be along a single segment that does not adjoin the vertex, so we have air at the vertex, and contemplate such a situation that has been produced by injection at a number of points into an initially empty cell. We show that, if we assume the pressure to be constant along the free boundaries, the region occupied by the fluid is the image of a rectangle under a conformal map that can be expressed in terms of elliptic functions if α = 1 or α = 2, and in terms of theta functions if 0 < α < 1 or 1 < α < 2. The form of the function giving the map can be written down, and the parameters appearing in it then determined as the solution to a set of transcendental equations. The procedure is illustrated by a number of examples.

Wave solutions of evolution equations and Hamiltonian flows on nonlinear subvarieties of generalized Jacobians

The algebraic-geometric approach is extended to study solutions of N-component systems associated with the energy dependent Schrodinger operators having potentials with poles in the spectral parameter, in connection with Hamiltonian flows on nonlinear subvariaties of Jacobi varieties. The systems under study include the shallow water equation and Dym type equation. The classes of solutions are described in terms of theta-functions and their singular limits by using new parameterizations. A qualitative description of real valued solutions is provided.

One-dimensional Linear Flow With Constant Terminal Pressures

Abstract A one-dimensional linear flow system of slight compressibility and of uniform properties is considered. At each boundary the pressure is constant, below the initial value, and generally different. At any time, the flow rate distribution within the system is found analytically in terms of theta functions, and the position of the drainage front is given in terms of an incomplete elliptic function. Except for equal terminal pressures, no proportionality exists between production rates and drainage areas, probably because the time to obtain boundary dominated flow is larger than the time at which one of the rates becomes zero. Pressure buildups following shutting in one of the two ends were obtained using the rate distribution during production and the unit step-function. Results provide insight into the significance of parameters obtained from well tests in the presence of open nearby wells. Introduction For wells producing at constant rates from a finite and homogeneous reservoir, the volume drained by each well is proportional to its production rate once pseudo-steady-state has been reached. This paper explores the relationship between drainage volumes and rates (or other parameters) for wells producing at constant bottom-hole pressure, for which no pseudo-steady-state can exist. Under the customary idealized conditions of homogeneity and uniformity of properties and single fluid flow, the drainage volumes have two unknowns: shape and size. The one-dimensional linear system chosen for analyses reduces the unknowns to one: the size of the drainage volume. In plan view, the shapes of the drainage volumes are rectangles. Rodriguez and Cinco-Ley(1) and Camacho et al.(2) have recently considered multi-well, multi-dimensional systems producing at constant bottom- hole pressure. Their results give rise to the investigation considered here. We are unaware of any prior investigation on this subject. Model The system considered is linear in x, of length L, and having a unit cross-sectional area normal to the x-direction. There are two plane sinks (wells), one at × = 0, the other at × = L, in a formation containing a single fluid having a small and constant compressibility c, and a mobility λ. The system has a uniform and homogeneous hydraulic diffusivity h, and is initially at a uniform pressure Differential Equation For the dimensionless variables defined in the Nomenclature, the differential equation and the initial and boundary conditions are Equation 1–4 (available in full paper) Sometimes it is advantageous to use a dimensionless time t defined by Equation 5 (available in full paper) Solution The solution to the differential Equation (1) subject to the initial and boundary conditions given by Equations (2) – (4) is given by Carslaw and Jaeger(3) to be Equation 6 (available in full paper) The pressure may then be manipulated to obtain additional results of interest. The flow rate anywhere in the reservoir, obtained from the spatial derivative of Equation (6) and the definitions of the theta functions, is Equation 7 (available in full paper) Other results are concisely defined in terms of a dimensionless parameter m, related to the dimensionless time by Equation 8 (available in full paper)

On billiard solutions of nonlinear PDEs

This Letter presents some special features of a class of integrable PDEs admitting billiard-type solutions, which set them apart from equations whose solutions are smooth, such as the KdV equation. These billiard solutions are weak solutions that are piecewise smooth and have first derivative discontinuities at peaks in their profiles. A connection is established between the peak locations and finite dimensional billiard systems moving inside n-dimensional quadrics under the field of Hooke potentials. Points of reflection are described in terms of theta-functions and are shown to correspond to the location of peak discontinuities in the PDEs weak solutions. The dynamics of the peaks is described in the context of the algebraic-geometric approach to integrable systems.

Central values of Hecke L-functions of CM number fields

0. Introduction. It is well known that the zeta function of CM (complex multiplication) abelian varieties can be given in terms of L-functions of associated Hecke characters. In this paper, we prove a formula expressing the central special value of the L-function of certain Hecke characters in terms of theta functions. The formula easily implies that the central value is nonnegative and yields a criterion for its positivity. Combining this criterion with the work of Arthaud and Rubin, we show that certain CM elliptic curves have Mordell-Weil rank zero over their field of definition. Let F be a totally real number field of degree t and let µ be a quadratic Hecke character of F of conductor f such that (2F , f) = 1. Given a CM extension E of F , we consider the twist χ = χcan ˜ µ of µ by a “canonical” Hecke character χcan of

Complete sets of Bloch and Wannier functions composed of oscillator eigenfunctions

We discuss Bloch and Wannier functions related to oscillator eigenfunctions. In particular, we construct complete sets of mutually orthogonal Bloch and Wannier functions. We show that they can be expressed in several ways in terms of theta functions and their derivatives. We also analyze their localization properties and discuss expectation values for specifically chosen Hamiltonians.

On the quintuple product identity

In this note we present a new proof of the quintuple product identity which is based on our study of k t h k^{th} order theta functions with characteristics and the identities they satisfy. In this context the quintuple product identity is another example of an identity which when phrased in terms of theta functions, rather than infinite products and sums, has a simpler form and is much less mysterious.

The Uses of Whitham Hierarchies

We review some of the uses of Whitham hierarchies in the context of the theory of the prepotential in N = 2 supersymmetric gauge theories. We focus on the structure of the contact terms in the twisted topological theory, and on the connection between Whitham hierarchies and the u-plane integrals for higher rank gauge groups, trying to put together the different approaches involved in this connection. We also review two other uses of the Whitham hierarchies: the interpretation of the slow times as supersymmetry breaking parameters, and the new techniques to extract instanton corrections using the RG equations written in terms of theta functions.

Resolutions and characters of irreducible representations of the N = 2 superconformal algebra

We evaluate characters of irreducible representations of the N = 2 supersymmetric extension of the Virasoro algebra. We do so by deriving the BGG resolution of the admissible N = 2 representations and also a new “3, 5, 7, …” resolution in terms of twisted massive Verma modules. We analyse how the characters behave under the automorphisms of the algebra, whose most significant part is the spectral flow transformations. The possibility to express the characters in terms of theta functions is determined by their behaviour under the spectral flow. We also derive the identity expressing every sl ̂ (2) character as a linear combination of spectral-flow transformed N = 2 characters; this identity involves a finite number of N = 2 characters in the case of unitary representations. Conversely, we find an integral representation for the admissible N = 2 characters as contour integrals of admissible sl ̂ (2) characters.

Explicit solution of Riemann-Hilbert problems for the Ernst equation

Riemann\char21{}Hilbert problems are an important solution technique for completely integrable differential equations. They are used to introduce a free function in the solutions which can be used at least in principle to solve initial or boundary value problems. But even if the initial or boundary data can be translated into a Riemann\char21{}Hilbert problem, it is in general impossible to obtain explicit solutions. In the case of the Ernst equation, however, this is possible for a large class because the matrix problem can be shown to be gauge equivalent to a scalar one on a hyperelliptic Riemann surface that can be solved in terms of theta functions. As an example we discuss the rigidly rotating dust disk.

On solutions of the Schlesinger equations in terms of theta-functions

The Schlesinger equations (see [18]) arise in the context of the following Riemann-Hilbert (inverse monodromy) problem: For an arbitrary g ∈ N and distinct 2g + 2 points λj ∈ C, construct a function Ψ(λ): CP1 {λ1, . . . , λ2g+2} → SL(2,C) which has the following properties: (1) Ψ(∞) = I; (2) Ψ(λ) is holomorphic for all λ ∈ CP1 {λ1, . . . , λ2g+2}; (3) Ψ(λ) has regular singular points at λ = λj, j = 1, . . . ,2g + 2, with given monodromy matrices, Mj ∈ SL(2,C). In the case where the monodromy matrices are independent of the parameters λ1, . . . , λ2g+2, the function Ψ ≡ Ψ(λ) solves the matrix differential equation

Inverse scattering as an infinite period limit

The purpose of this paper is to show how the well-known Dyson formula giving the solution of the KdV equation (or equivalently the solution of the inverse scattering problem for the Schrödinger operator) in the case of initial data with compact support, can be derived as an infinite period limit of the Its-Matveev formula which gives all periodic (and more) solutions of KdV (or. equivalently, solves the inverse Floquet-spectral problem) in terms of theta functions. This problem is motivated by the natural question of how to extend the known theory of KdV to cases other than those of decaying initial data on the one hand, and the algebro-geometric case on the other hand. We solve our problem by considering a local smooth potential which is periodized and letting the period go to infinity (at the level of the spectral problem). We use the Riemann-Hilbert formulation of the inverse spectral problem for the periodic case, and show how the Riemann-Hilbert problem corresponding to the inverse scattering theory for the initial local potential is recovered, as the period tends to infinity.