Algebraic-geometric Solutions Research Articles

Toda lattice with constraint of type B

We introduce a new integrable hierarchy of nonlinear differential–difference equations which is a subhierarchy of the 2D Toda lattice defined by imposing a constraint to the Lax operators of the latter. The 2D Toda lattice with the constraint can be regarded as a discretization of the BKP hierarchy. We construct its algebraic–geometrical solutions in terms of Riemann and Prym theta-functions.

Kadomtsev–Petviashvili Turning Points and CKP Hierarchy

A characterization of the Kadomtsev–Petviashvili hierarchy of type C (CKP) in terms of the KP tau-function is given. Namely, we prove that the CKP hierarchy can be identified with the restriction of odd times flows of the KP hierarchy on the locus of turning points of the second flow. The notion of CKP tau-function is clarified and connected with the KP tau function. Algebraic–geometrical solutions and in particular elliptic solutions are discussed in detail. A new identity for theta-functions of curves with holomorphic involution having fixed points is obtained.

On a Toda lattice hierarchy: Lax pair, integrable symplectic map and algebraic–geometric solution

A Toda lattice hierarchy is studied by introducing a new spectral problem which is a discrete counterpart of the generalized Kaup–Newell spectral problem. Based on the Lenard recursion equation, Lax pair of the hierarchy is given. Further, the discrete spectral problem is nonlinearized into an integrable symplectic map. As a result, an algebraic–geometric solution in Riemann theta function of the hierarchy is obtained. Besides, two equations, the Volterra lattice and a (2[Formula: see text]+[Formula: see text]1)-dimensional Burgers equation with a discrete variable, yielded from the hierarchy are also solved.

Algebraic-geometric solutions of the Dirac hierarchy

Введено уравнение Ленарда, приведены два его специальных решения. Первое решение используется для вывода расширенной иерархии Дирака, второе решение применяется для построения производящей функции. Производящая функция дает сохраняющиеся интегралы гамильтоновой системы Дирака и определяет алгебраическую кривую. На основе теории алгебраических кривых доказано, что гамильтонова система Дирака является интегрируемой, и получены алгебро-геометрические решения иерархии Дирака.

Two kinds of discrete integrable hierarchies of evolution equations and some algebraic-geometric solutions

With the help of a loop algebra we first present a (1+1)-dimensional discrete integrable hierarchy with a Hamiltonian structure and generate a (2+1)-dimensional discrete integrable hierarchy, respectively. Then we obtain a new differential-difference integrable system with three-potential functions, whose algebraic-geometric solution is derived from the theory of algebraic curves, where we construct the new elliptic coordinates to straighten out the continuous and discrete flows by introducing the Abel maps as well as the Riemann-Jacobi inversion theorem.

Algebraic-geometrical solutions of three (2 + 1)-dimensional nonlinear equations via Hamiltonian approach

The algebraic-geometrical solutions of three (2 + 1)-dimensional equations (including mKP equation and coupled mKP equation) are discussed by Hamiltonian approach. First, the Poisson structure on CN × RN is introduced to give a Hamiltonian system associated with the derivative nonlinear Schrödinger (DNLS) hierarchy. The Hamiltonian system is proved to be Liouville integrable, accordingly the solutions of three (2 + 1)-dimensional nonlinear equations can be solved by three compatible Hamiltonian flows. Second, the canonical separated variables and Hamilton–Jacobi theory is used to definite action-angle variables for Hamiltonian flows. At last, by Riemann–Jacobi inversion, the algebraic-geometrical solutions of three (2 + 1)-dimensional nonlinear equations are obtained. Besides, the algebraic-geometrical solutions of the first two DNLS equations are also given.

An alternative approach to solve the mixed AKNS equations

The algebraic–geometric solutions of the mixed AKNS equations are investigated through a finite-dimensional Lie–Poisson Hamiltonian system, which is generated by the nonlinearization of the adjoint equation related to the AKNS spectral problem. First, each mixed AKNS equation can be decomposed into two compatible Lie–Poisson Hamiltonian flows. Then the separated variables on the coadjoint orbit are introduced to study these Lie–Poisson Hamiltonian systems. Further, based on the Hamilton–Jacobi theory, the relationship between the action-angle coordinates and the Jacobi-inversion problem is established. In the end, using Riemann–Jacobi inversion, the algebraic–geometric solutions of the first three mixed AKNS equations are obtained.

Zero level of a purely magnetic two-dimensional nonrelativistic Pauli operator for SPIN-1/2 particles

We study the manifold of complex Bloch-Floquet eigenfunctions for the zero level of a two-dimensional nonrelativistic Pauli operator describing the propagation of a charged particle in a periodic magnetic field with zero flux through the elementary cell and a zero electric field. We study this manifold in full detail for a wide class of algebraic-geometric operators. In the nonzero flux case, the Pauli operator ground state was found by Aharonov and Casher for fields rapidly decreasing at infinity and by Dubrovin and Novikov for periodic fields. Algebraic-geometric operators were not previously known for fields with nonzero flux because the complex continuation of “magnetic” Bloch-Floquet eigenfunctions behaves wildly at infinity. We construct several nonsingular algebraic-geometric periodic fields (with zero flux through the elementary cell) corresponding to complex Riemann surfaces of genus zero. For higher genera, we construct periodic operators with interesting magnetic fields and with the Aharonov-Bohm phenomenon. Algebraic-geometric solutions of genus zero also generate soliton-like nonsingular magnetic fields whose flux through a disc of radius R is proportional to R (and diverges slowly as R → ∞). In this case, we find the most interesting ground states in the Hilbert space L 2 (ℝ 2 ).

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.

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.

Algebraic-Geometric Solution to (2+1)-Dimensional Sawada–Kotera Equation

Special solution of the (2+1)-dimensional Sawada-Kotera equation is decomposed into three (0+1)-dimensional Bargmann flows. They are straightened out on the Jacobi variety of the associated hyperelliptic curve. Explicit algebraic-geometric solution is obtained on the basis of a deeper understanding of the KdV hierarchy.

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.

Algebraic-geometrical solutions of some multidimensional nonlinear evolution equations

The known (2+1)-dimensional breaking soliton equation, the coupled KP equation with three potentials and a new (3+1)-dimensional nonlinear evolution equation are decomposed into systems of solvable ordinary differential equations with the help of the (1+1)-dimensional AKNS equations. The Abel–Jacobi coordinates are introduced to straighten out the associated flows, from which algebraic-geometrical solutions of the (2+1)-dimensional breaking soliton equation, the coupled KP equation and the (3+1)-dimensional evolution equation are explicitly given in terms of the Riemann theta functions.

Algebraic geometrical solutions forcertain evolution equations and Hamiltonian flows on nonlinearsubvarieties of generalized Jacobians

Algebraic geometrical solutions of a new shallow-water equationand Dym-type equation are studied in connection with Hamiltonianflows on nonlinear subvarieties of hyperelliptic Jacobians.These equations belong to a class of N-component integrablesystems generated by Lax equations with energy-dependentSchrödinger operators having poles in the spectral parameter.The classes of quasi-periodic and soliton-type solutions ofthese equations are described in terms of theta- andtau-functions by using new parametrizations. A qualitativedescription of real-valued solutions is provided.

The -Matrix and an Algebraic-Geometric Solution of the AKNS System

We construct an approach to finite-dimensional integrable systems with nonlinear evolution equations from the standpoint of the -matrix and an algebraic-geometric solution, illustrating the method with the well-known AKNS equation. We present the -matrix of the constrained AKNS flow and obtain the algebraic-geometric solution of the AKNS equation.

Algebraic-geometric solutions of the Krichever-Novikov equation

A zero-curvature representation with constant poles on an elliptic curve is obtained for the Krichever-Novikov equation. Algebraic-geometric solutions of this equation are constructed. The consideration is based on reducing the theta function of a two-sheet covering of an elliptic curve to the Prym theta functions of codimension one.

Algebraic-geometric solutions of the Harry Dym equation

Algebraic-geometric solutions of the Harry Dym equation

R-matrix and algebraic-geometric solution for integrable symplectic map

A new Lax matrix is introduced for the integrable symplectic map (ISM), and the non-dynamical (i.e. constant)r-matrix of ISM is obtained. Moreover, an effective approach is systematically presented to construct the explicit solution (here, the explicit solution means algebraic-geometric solution expressed by the Riemann-Theta function) of a soliton system or nonlinear evolution equation from Lax matrix,r-matrix, and the theory of nonlinearization through taking the Toda lattice as an example. The given algebraic-geometric solution of the Toda lattice is almost-periodic and includes the periodic and finite-band solution.

Periodic solitons and real algebrale curves

We describe a metbod of calculation of alt pbysical algebraicgeometrical solutions of KP-equations. 1. lix the middle of seventies ha works of 5. P. Novikov and others was discovered a new metitod of solution fon sorne important differential equations of matitematical pit>’sics [4]. At first, 1 sitorti>’ remind titis metitod. Jet B = {B 1~} E CL(y, 0)

Coin Tossing

The idea for this article came from a problem that was published before the widespread availability of graphing calculators (North Carolina School for Science and Mathematics 1988). In that publication, Geometric Probability, an interesting analytic solution to the problem about comparing the areas of squares was given and is described later in Algebraic-Geometric Solution. We have adapted the problem and have used successfully another, equally interesting method of solution in numerous classes from seventh-grade prealgebra through precalculus, as well as with several groups of teachers. This article presents our solution and, in addition, an example of how a rational function, a type not commonly found in applications of mathematics at this level, can model a solution of a related problem. The problem that we use with our students follows: