Jacobi Inversion Research Articles

VARIABLE SEPARATION AND ALGEBRAIC-GEOMETRIC SOLUTIONS OF MODIFIED TODA LATTICE EQUATION

In this paper, we derive a modified Toda lattice hierarchy by resorting to the Lenard operator pairs. The hyper-elliptic curve and Abel–Jacobi coordinates are then introduced to linearize the associated flow, from which some algebraic-geometric solutions of the modified Toda lattice are explicitly constructed in terms of Riemann theta functions by using Jacobi inversion technique.

Hilbert problem for a multiply connected circular domain and the analysis of the Hall effect in a plate

In this paper we analyze the Hilbert boundary-value problem of the theory of analytic functions for an ( N + 1 ) (N+1) -connected circular domain. An exact series-form solution has already been derived for the case of continuous coefficients. Motivated by the study of the Hall effect in a multiply connected plate we extend these results by examining the case of discontinuous coefficients. The Hilbert problem maps into the Riemann-Hilbert problem for symmetric piece-wise meromorphic functions invariant with respect to a symmetric Schottky group. The solution to this problem is derived in terms of two analogues of the Cauchy kernel, quasiautomorphic and quasimultiplicative kernels. The former kernel is known for any symmetric Schottky group. We prove the existence theorem for the second (quasimultiplicative) kernel for any Schottky group (its series representation is known for the first class groups only). We also show that the use of an automorphic kernel requires the solution to the associated real analogue of the Jacobi inversion problem, which can be bypassed if we employ the quasiautomorphic and quasimultiplicative kernels. We apply this theory to a model steady-state problem on the motion of charged electrons in a plate with N + 1 N+1 circular holes with electrodes and dielectrics on the walls when the conductor is placed at a right angle to the applied magnetic field.

Higher Genus Abelian Functions Associated with Cyclic Trigonal Curves

We develop the theory of Abelian functions associated with cyclic trigonal curves by considering two new cases. We investigate curves of genus six and seven and consider whether it is the trigonal nature or the genus which dictates certain areas of the theory. We present solutions to the Jacobi inversion problem, sets of relations between the Abelian function, links to the Boussinesq equation and a new addition formula.

An integrable lattice equation related to the nKN system

The Lax pair of a lattice equation is given by a discrete spectral problem and the negative Kaup–Newell spectral problem. Based on the Lax nonlinearization technique, they are transformed into a symplectic map and a Hamiltonian system, which are integrable in the Liouville sense and are straightened out in the Jacobi coordinates. An algebraic–geometric solution of the lattice equation is obtained by the Riemann–Jacobi inversion. Explicit solutions of the associated lattice hierarchy are given.

Bethe Ansatz, Inverse Scattering Transform and Tropical Riemann Theta Function in a Periodic Soliton Cellular Automaton for An(1)

We study an integrable vertex model with a periodic boundary condition associated with U_q(A_n^{(1)}) at the crystallizing point q=0. It is an (n+1)-state cellular automaton describing the factorized scattering of solitons. The dynamics originates in the commuting family of fusion transfer matrices and generalizes the ultradiscrete Toda/KP flow corresponding to the periodic box-ball system. Combining Bethe ansatz and crystal theory in quantum group, we develop an inverse scattering/spectral formalism and solve the initial value problem based on several conjectures. The action-angle variables are constructed representing the amplitudes and phases of solitons. By the direct and inverse scattering maps, separation of variables into solitons is achieved and nonlinear dynamics is transformed into a straight motion on a tropical analogue of the Jacobi variety. We decompose the level set into connected components under the commuting family of time evolutions and identify each of them with the set of integer points on a torus. The weight multiplicity formula derived from the q=0 Bethe equation acquires an elegant interpretation as the volume of the phase space expressed by the size and multiplicity of these tori. The dynamical period is determined as an explicit arithmetical function of the n-tuple of Young diagrams specifying the level set. The inverse map, i.e., tropical Jacobi inversion is expressed in terms of a tropical Riemann theta function associated with the Bethe ansatz data. As an application, time average of some local variable is calculated.

Abelian functions associated with a cyclic tetragonal curve of genus six

We develop the theory of Abelian functions defined using a tetragonal curve of genus six, discussing in detail the cyclic curve y4 = x5 + λ4x4 + λ3x3 + λ2x2 + λ1x + λ0. We construct Abelian functions using the multivariate σ-function associated with the curve, generalizing the theory of the Weierstrass ℘-function. We demonstrate that such functions can give a solution to the KP-equation, outlining how a general class of solutions could be generated using a wider class of curves. We also present the associated partial differential equations satisfied by the functions, the solution of the Jacobi inversion problem, a power series expansion for σ(u) and a new addition formula.

The positive and negative Camassa–Holm-γ hierarchies, zero curvature representations, bi-Hamiltonian structures, and algebro-geometric solutions

In this paper, we extend the Camassa–Holm-γ (CH-γ) equation to two different hierarchies of nonlinear evolution equations by resorting to two Lenard recursion sequences. One is of positive order CH-γ hierarchy including a Harry–Dym-type equation, while the other is of negative order CH-γ hierarchy of integrodifferential equations with nonzero integration constants including the CH-γ equation. We provide the zero curvature representations for both hierarchies by solving a key matrix equation. Moreover, the Lenard sequences are used to construct a Lax matrix that satisfies a stationary zero curvature equation, which enables us to establish bi-Hamiltonian structures for both hierarchies by applying trace identity. In addition, the Lenard sequence and Lax pair are used to systematically derive the Its–Matveev trace formula and the Dubrovin-type equations associated with the CH-γ equation. The hyperelliptic curve and Abel–Jacobi coordinates are then introduced to linearize the associated flow, from which the algebro-geometric solutions to the CH-γ equation are constructed by using standard Jacobi inversion technique.

Analytic solutions of the geodesic equation in higher dimensional static spherically symmetric spacetimes

The complete analytical solutions of the geodesic equation of massive test particles in higher dimensional Schwarzschild, Schwarzschild--(anti)de Sitter, Reissner-Nordstr\"om and Reissner--Nordstr\"om--(anti)de Sitter spacetimes are presented. Using the Jacobi inversion problem restricted to the theta divisor the explicit solution is given in terms of Kleinian sigma functions. The derived orbits depend on the structure of the roots of the characteristic polynomials which depend on the particle's energy and angular momentum, on the mass and the charge of the gravitational source, and the cosmological constant. We discuss the general structure of the orbits and show that due to the specific dimension-independent form of the angular momentum and the cosmological force a rich variety of orbits can emerge only in four and five dimensions. We present explicit analytical solutions for orbits up to 11 dimensions. A particular feature of Reissner-Nordstr\"om spacetimes is that bound and escape orbits traverse through different universes.

Hyperelliptic functions and geodesic equations

AbstractA method for solving geodesic equations in Schwarzschild–de Sitter space–times and higher dimensional Schwarzschild space–times is presented. The solutions are derived from Jacobi's inversion problem on a Riemann surface of genus 2 restricted to the set of zeros of the theta function, which is called a theta–divisor. In its final form, the solutions are given in terms of derivatives of Kleinian sigma functions. (© 2008 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)

Jacobi inversion on strata of the Jacobian of the C rs curve y r =f(x)

By using the generalized sigma function of a C rs curve y r =f(x) , we give a solution to the Jacobi inversion problem over the stratification in the Jacobian given by the Abel image of the symmetric products of the curve. We show that determinants consisting of algebraic functions on the curve, whose zeros give the Abelian pre-image of the strata, are written by ratios of certain derivatives of the sigma function on the strata. We also discuss the order of vanishing of abelian functions on the strata in terms of intersection theory.

Geodesic equation in Schwarzschild-(anti-)de Sitter space-times: Analytical solutions and applications

The complete set of analytic solutions of the geodesic equation in a Schwarzschild-(anti-)de Sitter space-time is presented. The solutions are derived from the Jacobi inversion problem restricted to the set of zeros of the theta function, called the theta divisor. In its final form the solutions can be expressed in terms of derivatives of Kleinian sigma functions. The different types of the resulting orbits are characterized in terms of the conserved energy and angular momentum as well as the cosmological constant. Using the analytical solution, the question whether the cosmological constant could be a cause of the Pioneer anomaly is addressed. The periastron shift and its post-Schwarzschild limit is derived. The developed method can also be applied to the geodesic equation in higher dimensional Schwarzschild space-times.

Complete Analytic Solution of the Geodesic Equation in Schwarzschild–(Anti-)de Sitter Spacetimes

The complete set of analytic solutions of the geodesic equation in a Schwarzschild-(anti-)de Sitter space-time is presented. The solutions are derived from the Jacobi inversion problem restricted to the theta divisor. In its final form the solutions can be expressed in terms of derivatives of Kleinian sigma functions. The solutions are completely classified by the structure of the zeros of the characteristic polynomial which depends on the energy, angular momentum, and the cosmological constant.

A (2+1)-dimensional derivative Toda equation in the context of the Kaup-Newell spectral problem

A (2+1)-dimensional derivative Toda equation is derived from the Lax triad composed of the Kaup–Newell, the negative Kaup–Newell and a discrete spectral problem. Two integrable Hamiltonian systems and an integrable symplectic map are derived from the Lax triad. They are straightened out in the Jacobi variety of the associated hyperelliptic curve. A finite genus solution is obtained through a technique based on the Riemann–Jacobi inversion theorem. Besides, explicit solutions are calculated for other associated integrable models, including the mKP equation and the nKN equation.

Quasi-periodic solutions for modified Toda lattice equation

Based on a spectral problem and the Lenard operator pairs, we derive in this paper a modified Toda lattice hierarchy. The modified Toda lattice equation is first decomposed into systems of integrable ordinary differential equations. A hyper-elliptic Riemann surface and Abel–Jacobi coordinates are then introduced to linearize the associated flow, from which some quasi-periodic solutions of the modified Toda lattice can be explicitly constructed in terms of Riemann theta functions by using Jacobi inversion technique.

A new hierarchy of (1 + 1)-dimensional soliton equations and its quasi-periodic solutions

A new spectral problem is proposed, from which a hierarchy of (1 + 1)-dimensional soliton equations is derived. With the help of nonlinearization approach, the soliton systems in the hierarchy are decomposed into two new compatible Hamiltonian systems of ordinary differential equations. The generating function flow method is used to prove the involutivity and the functional independence of the conserved integrals. The Abel–Jacobi coordinates are introduced to straighten out the associated flows. Using the Riemann–Jacobi inversion technique, the explicit quasi-periodic solutions for the (1 + 1)-dimensional soliton equations are obtained.

Method of Riemann surfaces in the study of supercavitating flow around two hydrofoils in a channel

In the framework of the Tulin model of supercavitating flow, the problem of reconstructing the free surface of a channel and the shapes of the cavities behind two hydrofoils placed in an ideal fluid is solved in closed form. The conformal map that transforms a parametric plane with three cuts along the real axis into the triple-connected flow domain is found by quadratures. The use of the theory of Riemann surfaces (the Schottky doubles) enables the non-linear model problem to be reduced to two separate Riemann–Hilbert problems on a hyperelliptic surface of genus two. The solution to the first problem is a rational function with certain zeros and poles on a Riemann surface. The second problem is solved in terms of singular integrals with the Weierstrass kernel. The essential singularities of the solution at the infinite points of the surface due to a pole of the kernel are removed by solving a real analogue of the Jacobi inversion problem on the surface. The unknown parameters of the conformal map are recovered from a system of certain algebraic and transcendental equations.

An integrable ()-dimensional Toda equation with two discrete variables

Abstract An integrable ( 2 + 1 )-dimensional Toda equation with two discrete variables is presented from the compatible condition of a Lax triad composed of the ZS–AKNS (Zakharov, Shabat; Ablowitz, Kaup, Newell, Segur) eigenvalue problem and two discrete spectral problems. Through the nonlinearization technique, the Lax triad is transformed into a Hamiltonian system and two symplectic maps, respectively, which are integrable in the Liouville sense, sharing the same set of integrals, functionally independent and involutive with each other. In the Jacobi variety of the associated algebraic curve, both the continuous and the discrete flows are straightened out by the Abel–Jacobi coordinates, and are integrated by quadratures. An explicit algebraic–geometric solution in the original variable is obtained by the Riemann–Jacobi inversion.

Algebro-Geometric Solution to Two New (2+1)-DimensionalModified Kadomtsev–Petviashvili Equations

Two new (2+1)-dimensional modified Kadomtsev–Petviashvili (mKP)equations are presented, which are related to a hierarchy of(1+1)-dimensional soliton equations. Through the nonlinearization of Laxpair and the Riemann–Jacobi inversion technique, the algebro-geometricsolutions of both the mKP equations are obtained.

Relaxation of nonlinear oscillations in BCS superconductivity

The diagonal case of the sl(2) Richardson–Gaudin quantum pairing model is known to be solvable as an Abel–Jacobi inversion problem. The effect of random time-dependent perturbations of the single-particle spectrum on the exact solution is an open question of considerable physical relevance. Weak perturbations introduce a new, slow time scale, while preserving the nonlinear character of the dynamics. In this paper, such perturbations are considered. It is shown that the long-time asymptotics can be obtained by a deformation of the original integrable system, equivalent to phase averaging over the fast time scale.

Some quasi-periodic solutions to the Kadometsev-Petviashvili and modified Kadometsev-Petviashvili equations

The Kadometsev-Petviashvili (KP) and modified KP (mKP) equations are retrieved from the first two soliton equations of coupled Korteweg-de Vries (cKdV) hierarchy. Based on the nonlinearization of Lax pairs, the KP and mKP equations are ultimately reduced to integrable finite-dimensional Hamiltonian systems in view of the r-matrix theory. Finally, the resulting Hamiltonian flows are linearized in Abel-Jacobi coordinates, such that some specially explicit quasi-periodic solutions to the KP and mKP equations are synchronously given in terms of theta functions through the Jacobi inversion.