Algebro-geometric Method Research Articles

Rogue-wave and breather solutions of the Fokas–Lenells equation on theta-function backgrounds

A systematic method is developed to obtain localized wave solutions of the Fokas–Lenells equation on theta-function backgrounds. First, using the properties of the theta-functions, we find a theta-function seed solution of the Fokas–Lenells equation. Next, based on the Riccati equation of Lax pair, we construct a Darboux transformation of the Fokas–Lenells equation. Then, the Kaup–Newell type spectral problem with theta-function potentials is solved by using the Baker–Akhiezer functions in the algebro-geometric method. Finally, Exact rogue-wave and breather solutions of the Fokas–Lenells equation on theta-function backgrounds are constructed by using the derived Darboux transformation and Baker–Akhiezer functions. In addition, the interaction dynamics of various localized wave solutions are analyzed by choosing appropriate parameters.

The algebro-geometric method: Solving algebraic differential equations by parametrizations

We present a survey of the algebro-geometric method for solving algebraic ordinary differential equations by means of parametrizations of the associated algebraic sets. In particular, we deal with equations of order one, and also systems of algebro-geometric dimension one. Various classes of solutions are treated symbolically, such as rational, algebraic, and power series solutions. We also consider classes of algebraic transformations of the associated algebraic sets preserving the solutions of the differential equations. Two Maple packages, implementing some of these solution methods, are presented.

Loci of 3-periodics in an Elliptic Billiard: Why so many ellipses?

A triangle center such as the incenter, barycenter, etc., is specified by a function thrice- and cyclically applied on sidelengths and/or angles. Consider the 1d family of 3-periodics in the elliptic billiard, and the loci of its triangle centers. Some will sweep ellipses, and others higher-degree algebraic curves. We propose two rigorous methods to prove if the locus of a given center is an ellipse: one based on computer algebra, and another based on an algebro-geometric method. We also prove that if the triangle center function is rational on sidelengths, the locus is algebraic.

Data Transmission Based on Exact Inverse Periodic Nonlinear Fourier Transform, Part I: Theory

The nonlinear Fourier transform (NFT) decomposes waveforms propagating through optical fiber into nonlinear degrees of freedom, which are preserved during transmission. By encoding information on the nonlinear spectrum, a transmission scheme inherently compatible with the nonlinear fiber is obtained. Despite potential advantages, the periodic NFT (PNFT) has been studied less compared to its counterpart based on vanishing boundary conditions, due to the mathematical complexity of the inverse transform. In this paper we extract the theory of the algebrogeometric integration method underlying the inverse PNFT from the literature, and tailor it to the communication problem. We provide a complete algorithm to compute the inverse PNFT. As an application, we employ the algorithm to design a novel modulation scheme called nonlinear frequency amplitude modulation, where four different nonlinear frequencies are modulated independently. Finally we provide two further modulation schemes that may be considered in future research. The algorithm is further applied in Part II of this paper to the design of a PNFT-based communication experiment.

Finite gap integration of the derivative nonlinear Schrödinger equation: A Riemann–Hilbert method

In this paper we retrieve finite gap solutions of the Gerdjikov–Ivanov type derivative nonlinear Schrödinger equation by using the algebro-geometric method and the Riemann–Hilbert method. We show that the Baker–Akhiezer function of the derivative nonlinear Schrödinger equation can be described in terms of two solvable matrix Riemann–Hilbert problems on ℂ with σ2- and σ3-symmetry conditions based on the technique developed in the study of long-time asymptotics. Our main tools include matrix Baker–Akhiezer function, asymptotic analysis, algebraic curve and Riemann theta function, matrix Riemann–Hilbert problem and associated deformation procedures.

Classification of meromorphic integrals for autonomous nonlinear ordinary differential equations with two dominant monomials

The Nevanlinna theory is used to derive the general structure of transcendental meromorphic solutions for a wide class of autonomous nonlinear ordinary differential equations with two dominant monomials. An algebro-geometric method, which enables one to obtain these solutions explicitly, is described. New simply periodic solutions of the Lorenz system are obtained.

The Algebro-Geometric Method for Solving Algebraic Differential Equations — A Survey

This paper presents the algebro-geometric method for computing explicit formula solutions for algebraic differential equations (ADEs). An algebraic differential equation is a polynomial relation between a function, some of its partial derivatives, and the variables in which the function is defined. Regarding all these quantities as unrelated variables, the polynomial relation leads to an algebraic relation defining a hypersurface on which the solution is to be found. A solution in a certain class of functions, such as rational or algebraic functions, determines a parametrization of the hypersurface in this class. So in the algebro-geometric method the author first decides whether a given ADE can be parametrized with functions from a given class; and in the second step the author tries to transform a parametrization into one respecting also the differential conditions. This approach is relatively well understood for rational and algebraic solutions of single algebraic ordinary differential equations (AODEs). First steps are taken in a generalization to systems and to partial differential equations.

Quasiperiodic Solutions of the Heisenberg Ferromagnet Hierarchy

We present quasi-periodic solutions in terms of Riemann theta functions of the Heisenberg ferromagnet hierarchy by using algebro-geometric method. Our main tools include algebraic curve and Riemann surface, polynomial recursive formulation and a special meromorphic function.

Quantum Compression Relative to a Set of Measurements

In this work, we investigate the possibility of compressing a quantum system to one of smaller dimension in a way that preserves the measurement statistics of a given set of observables. In this process, we allow for an arbitrary amount of classical side information. We find that the latter can be bounded, which implies that the minimal compression dimension is stable in the sense that it cannot be decreased by allowing for small errors. Various bounds on the minimal compression dimension are proven and an SDP-based algorithm for its computation is provided. The results are based on two independent approaches: an operator algebraic method using a fixed point result by Arveson and an algebro-geometric method that relies on irreducible polynomials and B\'ezout's theorem. The latter approach allows lifting the results from the single copy level to the case of multiple copies and from completely positive to merely positive maps.

A basis construction of the extended Catalan and Shi arrangements of the type A2

In [10], Terao proved the freeness of multi-Coxeter arrangements with constant multiplicities by giving an explicit construction of bases. Combining it with algebro-geometric method, Yoshinaga proved the freeness of the extended Catalan or Shi arrangements in [12]. However, there have been no explicit constructions of the bases for the logarithmic derivation modules of the extended Catalan and Shi arrangements. In this paper, we give the first explicit construction of them when the root system is of the type A2.

Partition a Quantum Pure-state Set into Unambiguously Discriminable Subsets

The problem of partitioning a quantum pure-state set into unambiguously discriminable subsets is investigated in this paper. We prove that the problem is decomposable into sub-problems of two-subset partitioning. A recursive algebro-geometric method based on Generalized Principal Component Analysis(GPCA) is proposed. When multiple feasible partitions exist, our method is able to return all of them by introducing Brill’s equations. Through this method, the efficiency of a kind of collective photon-number-splitting(PNS) attack is discussed.

Algebraic geometry and stability for integrable systems

In 1970s, a method was developed for integration of nonlinear equations by means of algebraic geometry. Starting from a Lax representation with spectral parameter, the algebro-geometric method allows to solve the system explicitly in terms of theta functions of Riemann surfaces. However, the explicit formulas obtained in this way fail to answer qualitative questions such as whether a given singular solution is stable or not. In the present paper, the problem of stability for equilibrium points is considered, and it is shown that this problem can also be approached by means of algebraic geometry.

Tropical Krichever construction for the non-periodic box and ball system

A solution for an initial value problem of the box and ball system is constructed from a solution of the periodic box and ball system. The construction is done through a specific limiting process based on the theory of tropical geometry. This method gives a tropical analogue of the Krichever construction, which is an algebro-geometric method to construct exact solutions to integrable systems, for the non-periodic system.

Integrable 1D Toda cellular automata

First, we recall the algebro-geometric method of construction of finite field valued solutions of the discrete KP equation (T i — shift operator in n i variable) and next we perform a reduction of dKP to the discrete 1D Toda equation This gives a method of construction of solutions of the discrete 1D Toda equation taking values in the finite field .

Algebro-Geometric Solution of the Discrete KP Equation over a Finite Field out of a Hyperelliptic Curve

We transfer the algebro-geometric method of construction of solutions of the discrete KP equation to the finite field case. We emphasize role of the Jacobian of the underlying algebraic curve in construction of the solutions. We illustrate in detail the procedure on example of a hyperelliptic curve.

The Hirota equation over finite fields: algebro-geometric approach and multisoliton solutions

We consider the Hirota equation (the discrete analog of the generalized Toda system) over a finite field. We present the general algebro-geometric method of construction of solutions of the equation. As an example we construct analogs of the multisoliton solutions for which the wave functions and the $\tau$-function can be found using rational functions. Within the class of multisoliton solutions we isolate generalized breather-type solutions which have no direct counterparts in the complex field case.

The Discrete KP and KdV Equations over Finite Fields

We propose an algebro-geometric method for constructing solutions of the discrete KP equation over a finite field. We also perform the corresponding reduction to the finite-field version of the discrete KdV equation. We write formulas that allow constructing multisoliton solutions of the equations starting from vacuum wave functions on an arbitrary nonsingular curve.

Character sums and intersection cohomology complexes associated to the space of square matrices

Let χ be a multiplicative character of a finite field F q . We study by an algebro-geometric method the finite Fourier transforms of χ(det x), where x runs over the square matrices over F q (untwisted case), or the Hermitian matrices (twisted case). We take up these examples as a first step to develop Intersection Cohomology Method in the theory of prehomogeneous vector spaces.