Hyperelliptic Theta Functions Research Articles

Dispersive shock wave, generalized Laguerre polynomials, and asymptotic solitons of the focusing nonlinear Schrödinger equation

We consider dispersive shock waves of the focusing nonlinear Schrödinger equation generated by discontinuous initial conditions which are periodic or quasiperiodic on the left semiaxis and zero on the right semiaxis. As an initial function, we use a finite-gap potential of the Dirac operator given in an explicit form through hyperelliptic theta-functions. The aim of this paper is to study the long-time asymptotics of the solution of this problem in a vicinity of the leading edge, where a train of asymptotic solitons are generated. Such a problem was studied in the work of Kotlyarov and Khruslov [Teor. Mat. Fiz. 68(2), 751–761 (1986)] and Kotlyarov {Mat. Zametki 49(2), 84–94 (1991) [Math. Notes 49(1-2), 172–180 (1991)]} using Marchenko’s inverse scattering techniques. We investigate this problem exceptionally using the Riemann-Hilbert (RH) problem techniques that allow us to obtain explicit formulas for asymptotic solitons themselves in contrast with the cited papers where asymptotic formulas are obtained only for the square of the absolute value of solution. Using transformations of the main RH problems, we arrive at a model problem corresponding to the parametrix at the end points of the continuous spectrum of the Zakharov-Shabat spectral problem. The parametrix problem is effectively solved in terms of the generalized Laguerre polynomials, which naturally appeared after appropriate scaling of the Riemann-Hilbert problem in small neighborhoods of the end points of the continuous spectrum. Further asymptotic analysis gives an explicit formula for solitons at the edge of dispersive waves. Thus, we give the complete description of the train of asymptotic solitons: not only bearing the envelope of each asymptotic soliton, but its oscillating structure is found explicitly. Besides, the second term of asymptotics describing an interaction between these solitons and oscillating background is also found. This gives the fine structure of the edge of dispersive shock waves.

The Global Parametrix in the Riemann-Hilbert Steepest Descent Analysis for Orthogonal Polynomials

In the application of the Deift-Zhou steepest descent method to the Riemann-Hilbert problem for orthogonal polynomials, a model Riemann-Hilbert problem that appears in the multi-cut case is solved with the use of hyperelliptic theta functions. We present here an alternative approach which uses meromorphic differentials instead of theta functions to construct the solution of the model Riemann-Hilbert problem. By using this representation, we obtain a new and elementary proof for the solvability of the model Riemann-Hilbert problem.

Hyperelliptic Theta-Functions and Spectral Methods: KdV and KP Solutions

This is the second in a series of papers on the numerical treatment of hyperelliptic theta-functions with spectral methods. A code for the numerical evaluation of solutions to the Ernst equation on hyperelliptic surfaces of genus 2 is extended to arbitrary genus and general position of the branch points. The use of spectral approximations allows for an efficient calculation of all characteristic quantities of the Riemann surface with high precision even in almost degenerate situations as in the solitonic limit where the branch points coincide pairwise. As an example we consider hyperelliptic solutions to the Kadomtsev–Petviashvili and the Korteweg–de Vries equations. Tests of the numerics using identities for periods on the Riemann surface and the differential equations are performed. It is shown that an accuracy of the order of machine precision can be achieved.

Hyperelliptic theta-functions and spectral methods

A code for the numerical evaluation of hyperelliptic theta-functions is presented. Characteristic quantities of the underlying Riemann surface such as its periods are determined with the help of spectral methods. The code is optimized for solutions of the Ernst equation where the branch points of the Riemann surface are parameterized by the physical coordinates. An exploration of the whole parameter space of the solution is thus only possible with an efficient code. The use of spectral approximations allows for an efficient calculation of all quantities in the solution with high precision. The case of almost degenerate Riemann surfaces is addressed. Tests of the numerics using identities for periods on the Riemann surface and integral identities for the Ernst potential and its derivatives are performed. It is shown that an accuracy of the order of machine precision can be achieved. These accurate solutions are used to provide boundary conditions for a code which solves the axisymmetric stationary Einstein equations. The resulting solution agrees with the theta-functional solution to very high precision.

On explicit solutions to the stationary axisymmetric Einstein‐Maxwell equations describing dust disks

AbstractWe review explicit solutions to the stationary axisymmetric Einstein‐Maxwell equations which can be interpreted as disks of charged dust. The disks of finite or infinite extension are infinitesimally thin and constitute a surface layer at the boundary of an electro‐vacuum. The Einstein‐Maxwell equations in the presence of one Killing vector are obtained by using a projection formalism. This leads to equations for three‐dimensional gravity where the matter is given by a SU(2,1)/S[U(1,1)× U(1)] nonlinear sigma model. The SU(2,1) invariance of the stationary Einstein‐Maxwell equations can be used to construct solutions for the electro‐vacuum from solutions to the pure vacuum case via a so‐called Harrison transformation. It is shown that the corresponding solutions will always have a non‐vanishing total charge and a gyromagnetic ratio of 2. Since the vacuum and the electro‐vacuum equations in the stationary axisymmetric case are completely integrable, large classes of solutions can be constructed with techniques from the theory of solitons. The richest class of physically interesting solutions to the pure vacuum case due to Korotkin is given in terms of hyperelliptic theta functions. Harrison transformed hyperelliptic solutions are discussed. As a concrete example we study the transformation of a family of counter‐rotating dust disks. To obtain algebro‐geometric solutions with vanishing total charge which are of astrophysical relevance, three‐sheeted surfaces have to be considered. The matter in the disk is discussed following Bičák et al. We review the ‘cut and glue’ technique where a strip is removed from an explicitly known spacetime and where the remainder is glued together after displacement. The discontinuities of the normal derivatives of the metric at the glueing hypersurface lead to infinite disks. If the energy conditions are satisfied and if the pressure is positive, the disks can be interpreted in the vacuum case as made up of two components of counter‐rotating dust moving on geodesics. In electro‐vacuum the condition of geodesic movement is replaced by electro‐geodesic movement. As an example we discuss a class of Harrison‐transformed hyperelliptic solutions. The range of parameters is identified where an interpretation of the matter in the disk in terms of electro‐dust can be given.

On explicit solutions to the stationary axisymmetric Einstein-Maxwell equations describing dust disks

We review explicit solutions to the stationary axisymmetric Einstein-Maxwell equations which can be interpreted as disks of charged dust. The disks of finite or infinite extension are infinitesimally thin and constitute a surface layer at the boundary of an electro-vacuum. The Einstein-Maxwell equations in the presence of one Killing vector are obtained by using a projection formalism. The SU(2,1) invariance of the stationary Einstein-Maxwell equations can be used to construct solutions for the electro-vacuum from solutions to the pure vacuum case via a so-called Harrison transformation. It is shown that the corresponding solutions will always have a non-vanishing total charge and a gyromagnetic ratio of 2. Since the vacuum and the electro-vacuum equations in the stationary axisymmetric case are completely integrable, large classes of solutions can be constructed with techniques from the theory of solitons. The richest class of physically interesting solutions to the pure vacuum case due to Korotkin is given in terms of hyperelliptic theta functions. The Harrison transformed hyperelliptic solutions are discussed.

Jacobian nullwerte and algebraic equations

We present two applications of jacobian nullwerte, both related with the resolution of algebraic equations of any degree. We give a very simple expression of the roots of a polynomial of arbitrary degree in terms of derivatives of hyperelliptic theta functions. This expression can be understood as an explicit proof of Torelli's theorem in the hyperelliptic case. We also give geometrical expressions of the discriminant of a polynomial. Both applications are based on a jacobian version of Thomae's formula.

Remarks on twisted theories with matter

We investigate some aspects of N=2 twisted theories with matter hypermultiplets in the fundamental representation of the gauge group. A consistent formulation of these theories on a general four-manifold requires turning on a particular magnetic flux, which we write down explicitly in the case of SU(2k). We obtain the blowup formula and show that the blowup function is given by a hyperelliptic sigma-function with singular characteristic. We compute the contact terms and find, as a corollary, interesting identities between hyperelliptic Theta functions.

Riemann-Hilbert problems for the Ernst equation and fibre bundles

Riemann-Hilbert techniques are used in the theory of completely integrable differential equations to generate solutions that contain a free function which can be used at least in principle to solve initial or boundary-value problems. The solution of a boundary-value problem is thus reduced to the identification of the jump data of the Riemann-Hilbert problem from the boundary data. But even if this can be achieved, it is very difficult to get explicit solutions since the matrix Riemann-Hilbert problem is equivalent to an integral equation. In the case of the Ernst equation (the stationary axisymmetric Einstein equations in vacuum), it was shown in a previous work that the matrix problem is gauge equivalent to a scalar problem on a Riemann surface. If the jump data of the original problem are rational functions, this surface will be compact which makes it possible to give explicit solutions in terms of hyperelliptic theta functions. In the present work, we discuss Riemann-Hilbert problems on Riemann surfaces in the framework of fibre bundles. This makes it possible to treat the compact and the non-compact case in the same setting and to apply general existence theorems.

Hyperelliptic Simple Factors of J0(N) with Dimension at Least 3

We develop algorithms for three problems. Starting with a complex torus of dimension g ≥ 2, isomorphic to a principally polarized, simple abelian variety A/C, the first problem is to find an algorithmic solution of the hyperelliptic Schottky problem: Is there a hvperelliptic curve C of genus g whose jacobian variety Ic is isomorphic to A over C? Our solution is based on [Poor 1994]. If such a hyperelliptic curve C exists, the next problem is the construction of the Rosenhain model C : y2 = X(X−1)(X−λ1)(X−λ2)…(X−λ2g−1) for pairwise distinct numbers λj E ∈ C \\ {0, 1}. Applying the theory of hyperelliptic theta functions we show that these numbers λj can easily be computed by using theta constants with even characteristics. If the abelian variety A is defined over a field k (this field could be the field of rational numbers, an algebraic number field of low degree, or a finite field), we show only in the case k = Q for simplicity, how the method in [Mestre 1991] can be generalized to get a minimal equation over Z[½] for the hyperelliptic curve C with jacobian variety Ic ≅c A. This is our third problem. For some hyperelliptic, principally polarized and simple factors with dimension g = 3, 4, 5 of the jacobian variety J0(N) = IX0(N) of the modular curve X0(N) we compute the corresponding curve equations by applying our algorithms to this special situation.

A characterization of “rapidity” curve in the chiral Potts model

We study the geometry of high genus curves of rapidity variables in chiral Potts model. In terms of symmetries, we characterize these Rieman surfaces and derive their period matrices. By the theory of prime forms, the temperature-like parameter is expressed by hyperelliptic theta functions.

A Generalization of the Weddle Surface, of Its Cremona Group, and of Its Parametric Expression in Terms of Hyperelliptic Theta Functions

A Generalization of the Weddle Surface, of Its Cremona Group, and of Its Parametric Expression in Terms of Hyperelliptic Theta Functions

An Addition Theorem for Hyperelliptic Theta-Functions

An Addition Theorem for Hyperelliptic Theta-Functions

Approximating tau-functions by theta-functions

We prove that the logarithm of an arbitrary tau-function of the KdV hierarchy can be approximated, in the topology of graded formal series by the logarithmic expansions of hyperelliptic theta-functions of finite genus, up to at most quadratic terms. As an example we consider theta-functional approximations of the Witten--Kontsevich tau-function.