arXiv subjects

Julia Bernatska

Publications and source records attributed to Julia Bernatska.

At least 19 recordsLinked to original sources

arXiv2026

Exact quasi-periodic solutions to the sine(sinh)-Gordon equations: The method for computation and analysis

The sine(sinh)-Gordon hierarchy of integrable Hamiltonian systems is described in detail, and all dynamic variables are expressed in terms of the $\wp$-functions that uniformize the associated spectral curve. Quasi-periodic solutions to the sine(sinh)-Gordon equations are obtained in terms of the function $\wp_{1,2g-1}$, reality conditions are revised, and a method of computation and analysis is presented. The proposed method is designed to analyze solutions by means of the Hamiltonian technique, which is illustrated in genera one and two.

nlin.SI↗

arXiv2025

Algebro-geometric integration of the Boussinesq hierarchy

We construct an integrable hierarchy of the Boussinesq equation using the Lie-algebraic approach of Holod-Flashka-Newell-Ratiu. We show that finite-gap hamiltonian systems of the hierarchy arise on coadjoint orbits in the loop algebra of $\mathfrak{sl}(3)$, and possess spectral curves from the family of $(3,3N\,{+}\,1)$-curves, $N\,{\in}\, \Natural$. Separation of variables leads to the Jacobi inversion problem on the mentioned curves, which is solved in terms of the corresponding multiply periodic functions. An exact finite-gap solution of the Boussinesq equation is obtained explicitly, and a conjecture on the reality conditions is made. The obtained solutions are computed for several spectral curves, and illustrated graphically.

nlin.SI↗

arXiv2025

Exact quasi-periodic solutions to the MKdV equation

In the present paper, a hierarchy of the mKdV equation is integrated by the methods of algebraic geometry. The mKdV hierarchy in question arises on coadjoint orbits in the loop algebra of $\mathfrak{sl}(2)$, and employs a family of hyperelliptic curves as spectral curves. A generic form of the finite-gap solution in any genus is obtained in terms of the $\wp$-functions, which generalize the Weierstrass $\wp$-function. Reality conditions for quasi-periodic wave solutions are completely specified. The obtained solutions are illustrated by plots in small genera.

nlin.SI↗

arXiv2025

Division polynomials in Mumford coordinates

An effective method of computing division polynomials in terms of Mumford coordinates is presented. As an example, division polynomials for $3$- and $4$-torsion divisors on a genus two curve are obtained explicitly in terms of Mumford coordinates, and $x$-, $y$-coordinates of the support of torsion divisors. As a result, $n$-torsion divisors on a given curve can be computed directly from the division polynomials. Alternatively, these divisors are obtained by solving the Jacobi inversion problem at points of the Jacobian variety of order $n$.

math.AG↗

arXiv2025

Abelian function fields on Jacobian varieties

In this paper the fields of multiply periodic, or Kleinian $\wp$-functions are exposed. Such a field arises on the Jacobian variety of an algebraic curve, and provides natural algebraic models of the Jacobian and Kummer varieties, possesses the addition law, and accommodates dynamical equations with solutions. All this will be explained in detail for plane algebraic curves in their canonical forms. Example of hyperelliptic and non-hyperelliptic curves are presented.

math.AG↗

arXiv2025

Computation of $\wp$-functions on plane algebraic curves

Numerical tools for computation of $\wp$-functions, also known as Kleinian, or multiply periodic, are proposed. In this connection, computation of periods of the both first and second kinds is reconsidered. An analytical approach to constructing Riemann surfaces of plane algebraic curves of low gonalities is used. The approach is based on explicit radical solutions to quadratic, cubic, and quartic equations, which serve for hyperelliptic, trigonal, and tetragonal curves, respectively. The proposed analytical models of Riemann surfaces give full control over computation of the Abel image of any point or divisor. Therefore, computation of $\wp$-functions at Abel images of given divisors can be done directly. An alternative computation with the help of the Jacobi inversion problem is used for verification. Hyperelliptic and trigonal curves are considered in detail, and illustrated by examples. A method of finding the unique characteristic corresponding to the vector of Riemann constants is suggested for non-hyperelliptic and hyperelliptic curves.

math-ph↗

arXiv2024

On Merton's Optimal Portfolio Problem with Sporadic Bankruptcy for Isoelastic Utility

We consider a stock that follows a geometric Brownian motion (GBM) and a riskless asset continuously compounded at a constant rate. We assume that the stock can go bankrupt, i.e., lose all of its value, at some exogenous random time (independent of the stock price) modeled as the first arrival time of a homogeneous Poisson process. For this setup, we study Merton's optimal portfolio problem consisting in maximizing the expected isoelastic utility of the total wealth at a given finite maturity time. We obtain an analytical solution using coupled Hamilton-Jacobi-Bellman (HJB) equations. The optimal strategy bans borrowing and never allocates more wealth into the stock than the classical Merton ratio recommends. For non-logarithmic isoelastic utilities, the optimal weights are non-myopic. This is an example where a realistic problem, being merely a slight modification of the usual GBM model, leads to non-myopic weights. For logarithmic utility, we additionally present an alternative derivation using a stochastic integral and verify that the weights obtained are identical to our first approach. We also present an example for our strategy applied to a stock with non-zero bankruptcy probability.

q-fin.MF↗

arXiv2024

Reality conditions for the KdV equation and exact quasi-periodic solutions in finite phase spaces

In the present paper reality conditions for quasi-periodic solutions of the KdV equation are determined completely. As a result, solutions in the form of non-linear waves can be plotted and investigated. The full scope of obtaining finite-gap solutions of the KdV equation is presented. It is proven that the multiply periodic $\wp_{1,1}$-function on the Jacobian variety of a hyperelliptic curve of arbitrary genus serves as the finite-gap solution, the genus coincides with the number of gaps. The subspace of the Jacobian variety where $\wp_{1,1}$, as well as other $\wp$-functions, are bounded and real-valued is found in any genus. This result covers every finite phase space of the KdV hierarchy, and can be extended to other completely integrable equations. A method of effective computation of this type of solutions is suggested, and illustrated in genera $2$ and $3$.

nlin.SI↗

arXiv2023

Solution of the Jacobi inversion problem on non-hyperelliptic curves

In this paper we propose a method of solving the Jacobi inversion problem in terms of multiply periodic $\wp$ functions, also called Kleinian $\wp$ functions. This result is based on the recently developed theory of multivariable sigma functions for $(n,s)$-curves. Considering $(n,s)$-curves as canonical representatives in the corresponding classes of bi-rationally equivalent plane algebraic curves, we claim that the Jacobi inversion problem on plane algebraic curves is solved completely. Explicit solutions on trigonal, tetragonal and pentagonal curves are given as an illustration.

math-ph↗

arXiv2022

Addition via reduction algorithm on trigonal curves

In this paper we propose a direct and explicit realization of addition of divisors by means of an iterative reduction algorithm. Each iteration of the algorithm is the reduction of a degree $g+1$ divisor to a divisor of degree~$g$. Such an approach allows to carry out all computations explicitly in a symbolic form, which is done for curves $C_{3,4}$, $C_{3,5}$ in this paper, and also for curves of higher genera up to $C_{3,13}$, $C_{3,14}$.

math.AG↗

arXiv2022

Solution of Mumford's second problem

A complete solution of Mumford's second problem about representation of theta derivatives with rational characteristics in terms of theta constants with rational characteristics is found. An explicit formula for computing such an expression for theta derivative with an arbitrary rational characteristic is derived, and illustrated with examples. Expressions for theta derivatives appear to be homogeneous of degree $3$ with respect to theta constants.

math.CV↗

arXiv2021

New generalisation of Jacobi's derivative formula

A stream of new theta relations is obtained. They follow from the general Thomae formula, which is a new result giving expressions for theta derivatives (the zero values of the lowest non-vanishing derivatives of theta functions with singular half-period characteristics) in terms of branch points and the period matrix of a hyperelliptic Riemann surface. The new theta relations contain (i) linear relations on the vector space of first order theta derivatives which are arranged in gradients, (ii) relations between second order theta derivatives and symmetric bilinear forms on the vector space of the gradients, (iii) relations between third order theta derivatives and symmetric trilinear forms on the vector space of the gradients, and (iv) a conjecture regarding higher order theta derivatives. It is shown how the Schottky identity (in the hyperelliptic case) is derived from the obtained relations.

math.AG↗

arXiv2020

General derivative Thomae formula for singular half-periods

The paper develops the result of second Thomae theorem in hyperelliptic case. The main formula, called general Thomae formula, provides expressions for values at zero of the lowest non-vanishing derivatives of theta functions with singular characteristics of arbitrary multiplicity in terms of branch points and period matrix. We call these values derivative theta constants. First and second Thomae formulas follow as particular cases. Some further results are derived. Matrices of second derivative theta constants (Hessian matrices of zero-values of theta functions with characteristics of multiplicity two) have rank three in any genus. Similar result about the structure of order $3$ tensor of third derivative theta constants is obtained, and a conjecture regarding higher multiplicities is made. As a byproduct a generalization of Bolza formulas are deduced.

math.AG↗

arXiv2020

Addition of Divisors on Hyperelliptic Curves via Interpolation Polynomials

Two problems are addressed: reduction of an arbitrary degree non-special divisor to the equivalent divisor of the degree equal to genus of a curve, and addition of divisors of arbitrary degrees. The hyperelliptic case is considered as the simplest model. Explicit formulas defining reduced divisors for some particular cases are found. The reduced divisors are obtained in the form of solution of the Jacobi inversion problem which provides the way of computing Abelian functions on arbitrary non-special divisors. An effective reduction algorithm is proposed, which has the advantage that it involves only arithmetic operations on polynomials. The proposed addition algorithm contains more details comparing with the known in cryptography, and is extended to divisors of arbitrary degrees comparing with the known in the theory of hyperelliptic functions.

math.AG↗

arXiv2019

Sato Grassmannian and Degenerate Sigma Function

The degeneration of the hyperelliptic sigma function is studied. We use the Sato Grassmannian for this purpose. A simple decomposition of a rational function gives a decomposition of Plücker coordinates of a frame of the Sato Grassmannian. It then gives a decomposition of the tau function corresponding to the degeneration of a hyperelliptic curve of genus $g$ in terms of the tau functions corresponding to a hyperelliptic curve of genus $g-1$. Since the tau functions are described by sigma functions, we get the corresponding formula for the degenerate hyperelliptic sigma function.

nlin.SI↗

arXiv2018

On Regularization of Second Kind Integrals

We obtain expressions for second kind integrals on non-hyperelliptic $(n,s)$-curves. Such a curve possesses a Weierstrass point at infinity which is a branch point where all sheets of the curve come together. The infinity serves as the basepoint for Abel's map, and the basepoint in the definition of the second kind integrals. We define second kind differentials as having a pole at the infinity, therefore the second kind integrals need to be regularized. We propose the regularization consistent with the structure of the field of Abelian functions on Jacobian of the curve. In this connection we introduce the notion of regularization constant, a uniquely defined free term in the expansion of the second kind integral over a local parameter in the vicinity of the infinity. This is a vector with components depending on parameters of the curve, the number of components is equal to genus of the curve. Presence of the term guarantees consistency of all relations between Abelian functions constructed with the help of the second kind integrals. We propose two methods of calculating the regularization constant, and obtain these constants for $(3,4)$, $(3,5)$, $(3,7)$, and $(4,5)$-curves. By the example of $(3,4)$-curve, we extend the proposed regularization to the case of second kind integrals with the pole at an arbitrary fixed point. Finally, we propose a scheme of obtaining addition formulas, where the second kind integrals, including the proper regularization constants, are used.

math.CV↗

arXiv2016

On degenerate sigma-functions of genus two

We obtain explicit expressions for genus 2 degenerate sigma-function in terms of genus $1$ sigma-function and elementary functions as solutions of a system of linear PDEs satisfied by the sigma-function. By way of application we derive a solution for a class of generalized Jacobi inversion problems on elliptic curves, a family of Schrödinger-type operators on a line with common spectrum consisting of a point and two segments, explicit construction of a field of three-periodic meromorphic functions. Generators of rank $3$ lattice in $\mathbb{C}^2$ are given explicitly.

math-ph↗

arXiv2014

A generalised Landau-Lifshitz equation for isotropic SU(3) magnet

In the paper we obtain equations for large-scale fluctuations of the mean field (the field of magnetization and quadrupole moments) in a magnetic system realized by a square (cubic) lattice of atoms with spin s >= 1 at each site. We use the generalized Heisenberg Hamiltonian with biquadratic exchange as a quantum model. A quantum thermodynamical averaging gives classical effective models, which are interpreted as Hamiltonian systems on coadjoint orbits of Lie group SU(3).

cond-mat.other↗