Hyperelliptic Solutions Research Articles

Maximal Amplitudes of Hyperelliptic Solutions of the Modified Nonlinear Schrödinger Equation

A simple formula is proven for the maximal amplitudes of N-phase hyperelliptic solutions of the modified nonlinear Schrödinger equation. The simple formula for the maximal amplitude depends only on the roots of an invariant polynomial.

Statistical mechanics of elastica for the shape of supercoiled DNA: Hyperelliptic elastica of genus three

This article studies the statistical mechanics of elastica as a model of the shapes of the supercoiled DNA, and shows that its excited states can be characterized by the focusing modified KdV (MKdV) equation due to thermal fluctuation. Following the previous paper (Matsutani and Previato, 2022), the hyperelliptic solutions of the focusing MKdV equation of genus three are considered. Numerical evaluation shows that the excited states given by the solution has a pattern as a repetition of the modulation of figure-eight and the inverse ‘S’ as a thermal fluctuation of elastica, called the S-eight mode. Our model states that the excited states of elastica due to the thermal effect have the S-eight mode. Since in the shapes of the AFM image of the supercoiled DNAs observed by Japaridze et al. (2017), the S-eight mode appears, our model is a novel model that mathematically represents the shape of the supercoiled DNA.

Maximal amplitudes of hyperelliptic solutions of the derivative nonlinear Schrödinger equation

AbstractA simple formula is proven for an upper bound for amplitudes of hyperelliptic (finite‐gap or N‐phase) solutions of the derivative nonlinear Schrödinger equation. The upper bound is sharp, viz, it is attained for some initial conditions. The method used to prove the upper bound is the same method, with necessary modifications, used to prove the corresponding bound for solutions of the focusing NLS equation (Wright OC, III. Sharp upper bound for amplitudes of hyperelliptic solutions of the focusing nonlinear Schrödinger equation. Nonlinearity. 2019;32:1929‐1966).

Sharp upper bound for amplitudes of hyperelliptic solutions of the focusing nonlinear Schrödinger equation

Hyperelliptic or finite-gap solutions of the focusing nonlinear Schrödinger equation are quasiperiodic N-phase solutions whose time dependence linearizes on a real subtorus of the Jacobi variety of an invariant hyperelliptic Riemann surface. A new proof is given for the formula for an upper bound, attained for some initial values, on amplitudes of hyperelliptic solutions. The upper bound is equal to the sum of the imaginary parts of all the branch points of the invariant Riemann surface which lie in the upper half-plane. A similar formula is proven for bounded solutions of the defocusing nonlinear Schrödinger equation. The new proof generalizes the derivation of the two-phase formula (Wright 2016 Physica D 321–2 16–38), and the result is the same N-phase formula obtained using Riemann–Hilbert methods by Bertola and Tovbis (2017 Commun. Math. Phys. 354 525–47).

On Reducible Degeneration of Hyperelliptic Curves and Soliton Solutions

In this paper we consider a reducible degeneration of a hyperelliptic curve of genus $g$. Using the Sato Grassmannian we show that the limits of hyperelliptic solutions of the KP-hierarchy exist and become soliton solutions of various types. We recover some results of Abenda who studied regular soliton solutions corresponding to a reducible rational curve obtained as a degeneration of a hyperelliptic curve. We study singular soliton solutions as well and clarify how the singularity structure of solutions is reflected in the matrices which determine soliton solutions.

Hyperelliptic function solutions with finite genus G of coupled nonlinear differential equations

In this paper, using the properties of hyperelliptic σ- and ℘- functions, , we propose an algorithm to obtain particular solutions of the coupled nonlinear differential equations, such as a general (2+1)- dimensional breaking soliton equation and static Veselov-Novikov(SVN) equation, the solutions of which can be expressed in terms of the hyperelliptic Kleinian functions for a given curve y 2=f(x) of (2g+1)- and (2g+2)- degree with genus G. In particular, owing to the idea of CK direct method, the algorithm can generate a series of new forms of hyperelliptic function solutions with the same genus G.

KdV–Volterra chain

This paper is devoted to the system of coupled KdV-like equations. It is shown that this apparently non-integrable system possesses an integrable reduction which is closely related to the Volterra chain. This fact is used to construct the hyperelliptic solutions of the original system.

Relations in a quantized elastica

In the previous article (Matsutani S 2002 J. Geom. Phys. 43 146–62), we showed the hyperelliptic solutions of a loop soliton as a study of a quantized elastica. Using the results, this paper studies relations between the quantized elastica and integrals of its Schwarz derivative, the winding effects in the quantized elastica problems and some other related relations.

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.

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.

Elliptic and hyperelliptic solutions of discrete Painlevé I and its extensions to higher order difference equations

The solutions of the discrete Painlevé equation I were constructed in terms of elliptic and hyperelliptic ψ functions for algebraic curves of genera one and two. For the case of genus two, there appear higher order difference equations which naturally contain the discrete Painlevé equation I as a special case.

Hyperelliptic solutions of modified Korteweg-de Vries equation of genus g: essentials of the Miura transformation

Explicit hyperelliptic solutions of the modified Korteweg-de Vries equations without any ambiguous parameters were constructed in terms only of the hyperelliptic $\al$-functions over non-degenerated hyperelliptic curve $y^2 = f(x)$ of arbitrary genus $g$. In the derivation, any $\theta$-functions or Baker-Akhiezer functions were not essentially used. Then the Miura transformation naturally appears as the connections between the hyperelliptic $\wp$-functions and hyperelliptic $\al$-functions.

Harrison transformation of hyperelliptic solutions and charged dust disks

We use a Harrison transformation on solutions to the stationary axisymmetric Einstein equations to generate solutions of the Einstein-Maxwell equations. The case of hyperelliptic solutions to the Ernst equation is studied in detail. Analytic expressions for the metric and the multipole moments are obtained. As an example we consider the transformation of a family of counter-rotating dust disks. The resulting solutions can be interpreted as disks with currents and matter with a purely azimuthal pressure or as two streams of freely moving charged particles. We discuss interesting limiting cases as the extreme limit where the charge becomes identical to the mass, and the ultrarelativistic limit where the central redshift diverges.

Hyperelliptic solutions of KdV and KP equations: re-evaluation of Baker's study on hyperelliptic sigma functions

Explicit function forms of hyperelliptic solutions of Korteweg-de Vries (KdV) and \break Kadomtsev-Petviashvili (KP) equations were constructed for a given curve $y^2 = f(x)$ whose genus is three. This study was based upon the fact that about one hundred years ago (Acta Math. (1903) {\bf{27}}, 135-156), H. F. Baker essentially derived KdV hierarchy and KP equation by using bilinear differential operator ${\bold{D}}$, identities of Pfaffians, symmetric functions, hyperelliptic $\sigma$-function and $\wp$-functions; $\wp_{\mu \nu} = -\partial_\mu \partial_\nu \log \sigma$ $= - ({\bold{D}}_\mu {\bold{D}}_\nu \sigma \sigma)/2\sigma^2$. The connection between his theory and the modern soliton theory was also discussed.

Higher‐order Painlevé Equations in the Polynomial Class I. Bureau Symbol P2

In this article, we construct all fourth‐ and fifth‐order differential equations in the polynomial class having the Painlevé property and having the Bureau symbol P2. The fourth‐order equations (including the Bureau barrier equation, y(iv)=3yy″−4(y′)2, which fails some Painlevé tests) are six in number and are denoted F‐I,…,F‐VI; the fifth‐order equations are four in number and are denoted Fif‐I,…,Fif‐IV. The 12 remaining equations of the fourth order in the polynomial class (where the Bureau symbol is P1) are listed in the Appendix, their proof of uniqueness being postponed to a sequel (paper II). Earlier work on this problem by Bureau, Exton, and Martynov is incomplete, Martynov having found 13 of the 17 distinct reduced equations. Equations F‐VI and Fif‐IV are new equations defining new higher‐order Painlevé transcendents. Other higher‐order transcendents appearing here may be obtained by group‐invariant reduction of the KdV5, Sawada–Kotera, and Kaup–Kupershmidt equations, the latter two being related. Four sections are devoted to solutions, first integrals, and assorted properties of the main equations. Several of the equations are solved in terms of hyperelliptic functions of genus 2 by means of Jacobi's postmultiplier theory. Except for a classic solution of Drach, we believe that all of these hyperelliptic solutions are new. In an accompanying paper, the hyperelliptic solutions of F‐V and F‐VI are applied to the unsolved third‐order Chazy classes IX and X.

Physically realistic solutions to the Ernst equation on hyperelliptic Riemann surfaces

We show that the class of hyperelliptic solutions to the Ernst equation (the stationary axisymmetric Einstein equations in vacuum) previously discovered by Korotkin and Meinel and Neugebauer can be derived via Riemann-Hilbert techniques. The present paper extends the discussion of the physical properties of these solutions that was begun in a previous Letter and supplies complete proofs. We identify a physically interesting subclass where the Ernst potential is everywhere regular except at a closed surface which might be identified with the surface of a body of revolution. The corresponding spacetimes are asymptotically flat and equatorially symmetric. This suggests that they could describe the exterior of an isolated body, for instance, a relativistic star or a galaxy. Within this class, one has the freedom to specify a real function and a set of complex parameters which can possibly be used to solve certain boundary value problems for the Ernst equation. The solutions can have ergoregions, a Minkowskian limit, and an ultrarelativistic limit where the metric approaches the extreme Kerr solution. We give explicit formulas for the potential on the axis and in the equatorial plane where the expressions simplify. Special attention is paid to the simplest nonstatic solutions (which are of genus 2) to which the rigidly rotating dust disk belongs.

Two-component reduction of Nahm's equations and hyper-elliptic solutions

We find the general hyper-elliptic solutions to the two-component reduced Nahm equations proposed by Hitchin et al. Elliptic solutions are a special case and can appear only for specific values of the monopole charges.