Terms Of Elliptic Functions Research Articles

Integrable cases in the dynamics of axial gyrostats and adiabatic invariants

This paper presents the study of axial gyrostats dynamics. The gyrostat is composed of two rigid bodies: an asymmetric platform and an axisymmetric rotor aligned with the platform principal axis. The paper discusses three types of gyrostats: oblate, prolate, and intermediate. Rotation of the rotor relative to the platform provides a source of small internal angular momentum and does not affect the moment of inertia tensor of the gyrostat. The dynamics of gyrostats without external torque is considered. The dynamics is described by using ordinary differential equations with Andoyer–Deprit canonical variables. For undisturbed motion, when the internal moment is equal to zero, the stationary solutions are found, and their stability is studied. General analytical solutions in terms of elliptic functions are also obtained. These results can be interpreted as the development of the classical Euler case for a solid, when to one degree of freedom—the relative rotation of bodies—is added. For disturbed motion of the gyrostats, when there is a system with slowly varying parameters, the adiabatic invariants are obtained in terms of complete elliptic integrals, which are approximately equal to the first integrals of the disturbed system. The adiabatic invariants remain approximately constant along a trajectory for long time intervals during which the parameter changes considerably. The results of the study can be useful for the analysis of dynamics of dual-spin spacecraft and for studying a chaotic behavior of the spacecraft.

Domain wall arrays, fronts, and bright and dark solitons in a generalized derivative nonlinear Schrödinger equation

We obtain some exact solutions of a generalized derivative nonlinear Schrödinger equation, including domain wall arrays (periodic solutions in terms of elliptic functions), fronts, and bright and dark solitons. In certain parameter domains, fundamental bright and dark solitons are chiral, and the propagation direction is determined by the sign of the self-steepening parameter. Moreover, we also find the chirping reversal phenomena of fronts, and bright and dark solitons, and discuss two different ways to produce the chirping reversal.

On Quispel-Roberts-Thompson extensions and integrable correspondences

We present several novel examples of an extension of the Quispel-Roberts-Thompson (QRT) mapping in which the coefficients are periodic functions of the independent variable and we prove that for period 2 this extension includes all so-called asymmetric QRT mappings. We establish the integrability of these new mappings by explicitly constructing rational invariants for them, which can be parametrised in terms of elliptic functions. Moreover, we show that the correspondences induced by these conserved quantities are all integrable – i.e., exhibit polynomial growth of the number of images they produce – and we present a conjecture pertaining to the degree of the growth. The case of Hirota-Kimura-Yahagi mappings and their associated correspondences is examined as well and we give examples of integrable correspondences associated with such mappings.

Chirped Domain Wall Arrays and Solitary Waves in a Generalized Nonlinear Schrödinger Equation

We study a generalized nonlinear Schrödinger equation, and obtain some exact solutions, including domain wall arrays (periodic solutions in terms of elliptic functions), fronts, bright and dark solitons. In certain parameter domains, fundamental bright and dark solitons show directionality and hence are chiral, and the propagation direction is determined by the sign of the self-steepening parameter. Moreover, the chirping reversal phenomena of bright and dark solitons are found.

Complete closed-form solutions of the Stark problem

Perturbed two-body problems play a special role in Celestial Mechanics as they capture the dominant dynamics for a broad range of natural and artificial satellites. In this paper, we investigate the classic Stark problem, corresponding to motion in a Newtonian gravitational field subjected to an additional uniform force of constant magnitude and direction. For both the two-dimensional and three-dimensional cases, the integrals of motion are determined, and the resulting quadratures are analytically integrated. A complete list of exact, closed-form solutions is deduced in terms of elliptic functions. It is found that all expressions rely on only seven fundamental solution forms. Particular attention is given to ensure that the expressions are well-behaved for very small perturbations. A comprehensive study of the phase space is also made using a boundary diagram to describe the domains of the general types of possible motion. Numerical examples are presented to validate the solutions.

Generalized QRT mappings with periodic coefficients

We present an extension of the QRT mapping beyond the familiar symmetric and asymmetric varieties. Starting from our results on discrete Painlevé equations, we show that there exist integrable QRT-like mappings, the coefficients of which are periodic functions. We present several examples of mappings with periodic coefficients of various periods and show that there exist cases where the periods are arbitrarily long. We prove the integrability of all the examples by constructing the corresponding conserved quantities and we show how these systems, just as their QRT siblings, can be explicitly integrated in terms of elliptic functions.

On the Hamiltonian structure of Hirota‐Kimura discretization of the Euler top

AbstractThis paper deals with a remarkable integrable discretization of the so (3) Euler top introduced by Hirota and Kimura. Such a discretization leads to an explicit map, whose integrability has been understood by finding two independent integrals of motion and a solution in terms of elliptic functions. Our goal is the construction of its Hamiltonian formulation. After giving a simplified and streamlined presentation of their results, we provide a bi‐Hamiltonian structure for this discretization, thus proving its integrability in the standard Liouville‐Arnold sense (© 2010 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)

On the Tetrahedrally Symmetric Monopole

We study SU(2) BPS monopoles with spectral curves of the form η 3+χ(ζ 6+b ζ 3−1) = 0. Previous work has established a countable family of solutions to Hitchin’s constraint that L 2 was trivial on such a curve. Here we establish that the only curves of this family that yield BPS monopoles correspond to tetrahedrally symmetric monopoles. We introduce several new techniques making use of a factorization theorem of Fay and Accola for theta functions, together with properties of the Humbert variety. The geometry leads us to a formulation purely in terms of elliptic functions. A more general conjecture than needed for the stated result is given.

New non-compact Calabi–Yau metrics in D = 6

A method for constructing explicit Calabi–Yau metrics in six dimensions in terms of an initial hyperkahler structure is presented. The equations to solve are nonlinear in general, but become linear when the objects describing the metric depend only on one complex coordinate of the hyperkahler four-dimensional space and its complex conjugate. This situation in particular gives a dual description of D6-branes wrapping a complex 1-cycle inside the hyperkahler space, Fayyazuddin (2007 Class. Quantum Grav. 24 3151). The present work generalizes the construction given in that reference. But the explicit solutions we present correspond to the nonlinear problem. This is a nonlinear equation with respect to two variables which, with the help of some specific anzatz, is reduced to a nonlinear equation with a single variable solvable in terms of elliptic functions. In these terms we construct an infinite family of non-compact Calabi–Yau metrics.

Single-Phase averaging for the Ablowitz-Ladik chain

The Bogolyubov-Whitham averaging method is applied to the Ablowitz-Ladik chain $$ \begin{gathered} - i\dot q_n - (1 - q_n r_n )(q_{n - 1} + q_{n + 1} ) + 2q_n = 0, \hfill \\ - i\dot r_n + (1 - q_n r_n )(r_{n - 1} + r_{n + 1} ) + 2r_n = 0 \hfill \\ \end{gathered} $$ in the single-phase case. We consider an averaged system and prove that the Hamiltonian property is preserved under averaging. The single-phase solutions are written in terms of elliptic functions and, in the “focusing” case, Riemannian invariants are obtained for modulation equations. The characteristic rates of the averaged system are stated in terms of complete elliptic integrals and the self-similar solutions of the systemare obtained. Results of the corresponding simulations are given.

Toward a Moyal quantization program of the membrane

A Moyal deformation quantization approach to a spherical membrane (moving in flat target backgrounds) in the light cone gauge is presented. The physical picture behind this construction relies on viewing the two spatial membrane coordinates σ1, σ2 as the two phase space variables q, p, and the temporal membrane coordinate τ as time. Solutions to the Moyal-deformed equations of motion are explicitly constructed in terms of elliptic functions. A knowledge of the Moyal-deformed light-cone membrane's Hamiltonian density H(q, p, τ) allows us to construct a time-dependent Wigner function ρ(q, p, τ) as solutions of the Moyal–Liouville equation, and from which one can obtain the expectation values of the operator that define the quantum average values of the energy density configurations of the membrane at any instant. It is shown how a time-dependent quartic oscillator with q4, p4, q2p2 terms plays a fundamental role in the quantum treatment of membranes and displays an important p↔q duality symmetry.

Exact computation of one-loop correction to the energy of folded spinning string in AdS5 × S5

We consider the one-loop correction to the energy of the folded spinning string solution in the AdS3 part of AdS5 × S5. The classical string solution is expressed in terms of elliptic functions so an explicit computation of the corresponding fluctuation determinants for generic values of the spin appears to be a non-trivial problem. We show how it can be solved exactly by using the static-gauge expression for the string partition function (which we demonstrate to be equivalent to the conformal gauge one) and observing that all the corresponding second-order fluctuation operators can be put into the standard (single-gap) Lamé form. We systematically derive the small-spin and large-spin expansions of the resulting expression for the string energy and comment on some of their applications.

Explicit peaked wave solutions to the generalized Camassa-Holm equation

By constructing auxiliary differential equations, we obtain peaked solitary wave solutions of the generalized Camassa-Holm equation, including periodic cusp waves expressed in terms of elliptic functions.

Geometric Sensitivity of a Pinhole Collimator

Geometric sensitivity for single photon emission computerized tomography (SPECT) is given by a double integral over the detection plane. It would be useful to be able to explicitly evaluate this quantity. This paper shows that the inner integral can be evaluated in the situation where there is no gamma ray penetration of the material surrounding the pinhole aperature. This is done by converting the integral to an integral in the complex plane and using Cauchy's theorem to replace it by one which can be evaluated in terms of elliptic functions.

Integrability of the discretization of the controlled Euler Top

The Euler equations with two controls are discretized by using the bilinear transformation method. The discrete equations are explicit and exhibit a sufficient number of conserved quantities for integrability. The explicit solution to the discrete controlled Euler top are deduced in terms of elliptic functions.

Use of Abel integral equations in water wave scattering by two surface-piercing barriers

In this paper the classical problem of water wave scattering by two partially immersed plane vertical barriers submerged in deep water up to the same depth is investigated. This problem has an exact but complicated solution and an approximate solution in the literature of linearised theory of water waves. Using the Havelock expansion for the water wave potential, the problem is reduced here to solving Abel integral equations having exact solutions. Utilising these solutions,two sets of expressions for the reflection and transmission coefficients are obtained in closed forms in terms of computable integrals in contrast to the results given in the literature which,involved six complicated integrals in terms of elliptic functions. The two different expressions for each coefficient produce almost the same numerical results although it has not been possible to prove their equivalence analytically. The reflection coefficient is depicted against the wave number in a number of figures which almost coincide with the figures available in the literature wherein the problem was solved approximately by employing complementary approximations. (C) 2009 Elsevier B.V. All rights reserved.

Explicit Solutions of (2+1)-Dimensional Canonical Generalized KP, KdV, and (2+1)-Dimensional Burgers Equations with Variable Coefficients

In this paper, using the generalized (G'/G)-expansion method and the auxiliary differential equation method, we discuss the (2+1)-dimensional canonical generalized KP (CGKP), KdV, and (2+1)-dimensional Burgers equations with variable coefficients. Many exact solutions of the equations are obtained in terms of elliptic functions, hyperbolic functions, trigonometric functions, and rational functions.

Evolution of extended clouds of particles near the sun and stars

The evolution of the trajectories of small objects not entirely ejected from the solar system and subject to significant perturbations owing to the regular galactic field is studied. These objects may be individual dust particles, comets, or, in the case of other stars, large dust clouds and stellar satellites. The possibility of averaging their motion over the period of the galactic rotation is examined. The averaged equations are solved in terms of elliptic functions. The eccentricity of the orbit also evolves, but returns periodically to its original value, ~1, corresponding to ejection if the interaction with individual stars is neglected. The slope of the line of apsides with respect to the galactic plane evolves in two ways: with time it either passes through 90° or through 0°.

Shocks and finite-time singularities in Hele-Shaw flow

Hele-Shaw flow at vanishing surface tension is ill-defined. In finite time, the flow develops cusp-like singularities. We show that this ill-defined problem admits a weak dispersive solution when singularities give rise to a graph of shock waves propagating into the viscous fluid. The graph of shocks grows and branches. Velocity and pressure have finite discontinuities across the shock. We formulate a few simple physical principles which single out the dispersive solution and interpret shocks as lines of decompressed fluid. We also formulate the dispersive weak solution in algebro-geometrical terms as an evolution of the Krichever–Boutroux complex curve. We study in detail the most generic (2, 3)-cusp singularity, which gives rise to an elementary branching event. This solution is self-similar and expressed in terms of elliptic functions.

Stability analysis of curvilinear configurations of an inextensible elastic rod with clamped ends

The paper addresses the stability problem of curvilinear configurations of a perfectly flexible rod known as elasticas. The question of stability of equilibrium configurations is reduced to investigating the sign of the energy functional in the presence of isoperimetric constraints. It is shown that the associated Jacobi equation can be integrated analytically, which makes it possible to perform the stability analysis for the rod equilibrium configurations using exact closed-form solutions expressed in terms of elliptic functions. The classical problem of a clamped–clamped rod is considered and complemented by the stability analysis.