Weierstrass Elliptic Function Research Articles

General complex envelope solutions of coupled-mode optics with quadratic or cubic nonlinearity

The analytic general solutions for the complex field envelopes are derived using Weierstrass elliptic functions for two and three mode systems of differential equations coupled via quadratic $\chi_2$ type nonlinearity as well as two mode systems coupled via cubic $\chi_3$ type nonlinearity. For the first time, a compact form of the solutions is given involving simple ratios of Weierstrass sigma functions (or equivalently Jacobi theta functions). A Fourier series is also given. All possible launch states are considered. The models describe sum and difference frequency generation, polarization dynamics, parity-time dynamics and optical processing applications.

Analytical solutions of the geodesic equation in the spacetime of a black hole inf(R)gravity

We consider the motion of test particles in the spacetime of a black hole in f(R) gravity. The complete set of analytic solutions of the geodesic equation in the spacetime of this black hole are presented. The geodesic equations are solved in terms of Weierstrass elliptic functions and 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 $\Lambda$ and the real constant $\beta$.

Optimal control of quantum systems with SU(1, 1) dynamical symmetry

SU(1, 1) dynamical symmetry is of fundamental importance in analyzing unbounded quantum systems in theoretical and applied physics. In this paper, we study the control of generalized coherent states associated with quantum systems with SU(1, 1) dynamical symmetry. Based on a pseudo Riemannian metric on the SU(1, 1) group, we obtain necessary conditions for minimizing the field fluence of controls that steer the system to the desired final state. Further analyses show that the candidate optimal control solutions can be classified into normal and abnormal extremals. The abnormal extremals can only be constant functions when the control Hamiltonian is non-parabolic, while the normal extremals can be expressed by Weierstrass elliptic functions according to the parabolicity of the control Hamiltonian. As a special case, the optimal control solution that maximally squeezes a generalized coherent state is a sinusoidal field, which is consistent with what is used in the laboratory.

Manifolds of solutions for Hirzebruch functional equations

For the nth Hirzebruch equation we introduce the notion of universal manifold Mn of formal solutions. It is shown that the manifold Mn, where n > 1, is algebraic and its dimension is not greater than n + 1. We give a family of polynomials generating the relation ideal in the polynomial ring on Mn. In the case n = 2 the generators of this ideal are described. As a corollary we obtain an effective description of the manifold M2 and therefore all series determining complex Hirzebruch genera that are fiberwise multiplicative on projectivizations of complex vector bundles. A family of analytic solutions of the second Hirzebruch equation is described in terms of Weierstrass elliptic functions and in terms of Baker–Akhiezer functions of elliptic curves. For this functions the curves differ, yet the series expansions in the vicinity of 0 coincide.

Characterization of piezoelectric composites with mechanical and electrical imperfect contacts

The aim of the present work is to study the influence of the mechanical and electrical imperfections in reinforced piezoelectric composite materials with unidirectional cylindrical fibers periodically distributed in rhombic cells under mechanical and electrical imperfect contacts. The behavior of the composites is studied through two approaches: the two-scale asymptotic homogenization method and the finite element method. The asymptotic homogenization method is applied to a two-phase composite with mechanical and electrical imperfect contacts and to a three-phase composite with perfect contact in the interphase. The constituents of the composite are homogeneous piezoelectric materials with transversely isotropic properties. The local problems are formulated for the spring-capacitor and three-phase models by the asymptotic homogenization method. The solution of each plane local problem is found using potential methods of a complex variable and the properties of doubly periodic Weierstrass elliptic functions. Closed-form formulae are obtained for the effective properties of the composites with both types of imperfect contacts and different configuration of the cells. The finite element method is implemented for analysis of piezoelectric composite materials with unidirectional cylindrical fibers periodically distributed in rhombic cells under mechanical and electrical imperfect contacts. Some numerical examples are given under the presence of both imperfect contacts and different arrangement of the cells. Comparisons between the numerical results reported by asymptotic homogenization method and finite element method are provided.

Dynamics of modulated waves in a lossy modified Noguchi electrical transmission line.

We study analytically the dynamics of modulated waves in a dissipative modified Noguchi nonlinear electrical network. In the continuum limit, we use the reductive perturbation method in the semidiscrete limit to establish that the propagation of modulated waves in the network is governed by a dissipative nonlinear Schrödinger (NLS) equation. Motivated with a solitary wave type of solution to the NLS equation, we use both the direct method and the Weierstrass's elliptic function method to present classes of bright, kink, and dark solitary wavelike solutions to the dissipative NLS equation of the network. Through the exact solitary wavelike solutions to the dissipative NLS equation, we investigate the effects of the dissipative elements of the network on wave propagation. We show that the wave amplitude decreases and its width increases when the dissipative element of the network increases. It has been also found that the dissipative element of the network can be used to manipulate the motion of solitary waves through the network. This work presents a good analytical approach of investigating the propagation of solitary waves through discrete electrical transmission lines and is very important for studying modulational instability.

The history of the Universe is an elliptic curve

Friedmann–Lemaître equations with contributions coming from matter, curvature, cosmological constant, and radiation, when written in terms of conformal time u rather than in terms of cosmic time t, can be solved explicitly in terms of standard Weierstrass elliptic functions. The spatial scale factor, the temperature, the densities, the Hubble function, and almost all quantities of cosmological interest (with the exception of t itself) are elliptic functions of u, in particular they are bi-periodic with respect to a lattice of the complex plane, when one takes u complex. After recalling the basics of the theory, we use these explicit expressions, as well as the experimental constraints on the present values of density parameters (we choose for the curvature density a small value in agreement with experimental bounds) to display the evolution of the main cosmological quantities for one real period of conformal time (the cosmic time t ‘never ends’ but it goes to infinity for a finite value of u). A given history of the Universe, specified by the measured values of present-day densities, is associated with a lattice in the complex plane, or with an elliptic curve, and therefore with two Weierstrass invariants . Using the same experimental data we calculate the values of these invariants, as well as the associated modular parameter and the corresponding Klein j-invariant. If one takes the flat case k = 0, the lattice is only defined up to homotheties, and if one, moreover, neglects the radiation contribution, the j-invariant vanishes and the corresponding modular parameter τ can be chosen in one corner of the standard fundamental domain of the modular group (equihanharmonic case: ). Several exact—i.e., non-numerical—results of independent interest are obtained in that case.

Analytical properties and exact solutions of the Lotka–Volterra competition system

The Lotka–Volterra competition system with diffusion is considered. The Painlevé property of this system is investigated. Exact traveling wave solutions of the Lotka–Volterra competition system are found. Periodic solutions expressed in terms of the Weierstrass elliptic function are also given.

On the extended Euler system and the Jacobi and Weierstrass elliptic functions

We study the extended Euler systems (EES) as an initial value problem. Particular realizations of it lead to several Lie-Poisson structures. We consider a 6-D Poisson structure that fit all of them together. The symplectic stratification of this non Lie-Poisson structure uses the first integrals which are elliptic and hyperbolic cylinders, although other quadrics may be used as well. A qualitative study of the solutions is carried out and the twelve Jacobi elliptic functions in the real domain are shown in an unified way as the solutions of the EES. As a consequence, Jacobi's transformation for the elliptic modulus is obtained. Likewise, introducing the square norm function we establish in a straightforward way the connection of the EES with the Weierstrass $\wp$ elliptic function, giving the relation of its invariants $g_i$ with the integrals and coefficients of the EES.

Lie Symmetry Analysis and New Exact Solutions for a Higher-Dimensional Shallow Water Wave Equation

In our work, a higher-dimensional shallow water wave equation, which can be reduced to the potential KdV equation, is discussed. By using the Lie symmetry analysis, all of the geometric vector fields of the equation are obtained; the symmetry reductions are also presented. Some new nonlinear wave solutions, involving differentiable arbitrary functions, expressed by Jacobi elliptic function, Weierstrass elliptic function, hyperbolic function, and trigonometric function are obtained. Our work extends pioneer results.

The periods of the generalized Jacobian of a complex elliptic curve

Abstract We show that the toroidal Lie group G = ℂ2/Λ, where Λ is the lattice generated by (1, 0), (0, 1) and (τ̂, τ͂), with τ̂ ∉ ℝ, is isomorphic to the generalized Jacobian JL of the complex elliptic curve C with modulus τ̂, defined by any divisor class L ≡ (M) + (N) of C fulfilling M − N = [℘ (τ͂) : ℘´(τ͂) : 1] ∈ C. This follows from an apparently new relation between the Weierstrass sigma and elliptic functions.

Полное решение уравнений Янга-Миллса для центрально-симметрической метрики при наличии электромагнитного поля

Ранее авторами было найдено полное решение уравнений Янга-Миллса для центрально-симметрической метрики в 4-мерном пространстве конформной связности без кручения при отсутствии электромагнитного поля. Позже, в другой статье, авторы нашли решения уравнений Янга-Миллса для этой же метрики при наличии электромагнитного поля специального вида, предполагая, что его компоненты зависят не от четырех, а только от двух переменных. Там же авторы провели сравнение получившегося решения с хорошо известным решением Райсснера-Нордстрема и указали причину того, что эти решения не совпадают. В настоящей работе авторы не накладывают никаких предварительных ограничений на компоненты электромагнитного поля. Это сильно усложняет вывод уравнений Янга-Миллса. Тем не менее, все вычислительные трудности были преодолены. Оказалось, что решения этих уравнений всe равно зависят только от двух переменных, и новых решений, кроме полученных ранее, не возникает. Авторы делают вывод, что найдены все решения уравнений Янга-Миллса для центрально-симметрической метрики при наличии произвольного электромагнитного поля, согласованного с уравнениями Янга-Миллса в пространстве без кручения (то есть без источников). Эти решения выражаются через эллиптическую функцию Вейерштрасса.

New Exact Solutions of the Davey–Stewartson Equation with Power-Law Nonlinearity

This work obtains the soliton solutions of the generalized Davey–Stewartson equation with the complex coefficients. First, the extended Weierstrass transformation method is used to carry out the solutions of this equation, and some new solutions, known as Weierstrass elliptic function solutions, are obtained by this method. Then, the trial equation method is used to obtain the soliton solutions of this equation.

New Exact Analytical Solutions for the General KdV Equation with Variable Coefficients

In this paper, a general algebraic method based on the generalized Jacobi elliptic functions expansion method, the improved general mapping deformation method and the extended auxiliary function method with computerized symbolic computation is proposed to construct more new exact solutions of a generalized KdV equation with variable coefficients. As a result, eight families of new generalized Jacobi elliptic function wave solutions and Weierstrass elliptic function solutions of the equation are obtained by using this method, some of these solutions are degenerated to soliton-like solutions, trigonometric function solutions in the limit cases when the modulus of the Jacobi elliptic functions m → 1 or 0, which shows that the general method is more powerful and will be used in further works to establish more entirely new solutions for other kinds of nonlinear partial differential equations arising in mathematical physics.

Some new addition formulae for Weierstrass elliptic functions.

We present new addition formulae for the Weierstrass functions associated with a general elliptic curve. We prove the structure of the formulae in n-variables and give the explicit addition formulae for the 2-and 3-variable cases. These new results were inspired by new addition formulae found in the case of an equianharmonic curve, which we can now observe as a specialization of the results here. The new formulae, and the techniques used to find them, also follow the recent work for the generalization of Weierstrass functions to curves of higher genus.

Comparison of three Stark problem solution techniques for the bounded case

Three methods of obtaining solutions to the Stark problem—one developed by Lantoine and Russell using Jacobi elliptic and related functions, one developed by Biscani and Izzo using Weierstrass elliptic and related functions, and one developed by Pellegrini, Russell, and Vittaldev using $$F$$ and $$G$$ Taylor series extended to the Stark problem—are compared qualitatively and quantitatively for the bounded motion case. For consistency with existing available code for the series solution, Fortran routines of the Lantoine method and Biscani method are newly implemented and made available. For these implementations, the Lantoine formulation is found to be more efficient than the Biscani formulation in the propagation of a single trajectory segment. However, for applications for which acceptable accuracy may be achieved by orders up to 16, the Pellegrini series solution is shown to be more efficient than either analytical method. The three methods are also compared in the propagation of sequentially connected trajectory segments in a low-thrust orbital transfer maneuver. Separate tests are conducted for discretizations between 8 and 96 segments per orbit. For the series solution, the interaction between order and step size leads to computation times that are nearly invariable to discretization for a given truncation error tolerance over the tested range of discretizations. This finding makes the series solution particularly attractive for mission design applications where problems may require both coarse and fine discretizations. Example applications include the modeling of low-thrust propulsion and time-varying perturbations—problems for which the efficient propagation of relatively short Stark segments is paramount because the disturbing acceleration generally varies continuously.

Non-Kaehler heterotic string solutions with non-zero fluxes and non-constant dilaton

Conformally compact and complete smooth solutions to the Strominger system with non vanishing flux, non-trivial instanton and non-constant dilaton using the first Pontrjagin form of the (-)-connection} on 6-dimensional non-Kaehler nilmanifold are presented. In the conformally compact case the dilaton is determined by the real slices of the elliptic Weierstrass function. The dilaton of non-compact complete solutions is given by the fundamental solution of the Laplacian on $R^4$.

Painlevé analysis and exact solutions for the Belousov–Zhabotinskii reaction–diffusion system

A system of equations for description of the Belousov–Zhabotinskii chemical reaction is considered. The Painlevé analysis of this reaction–diffusion system is studied. Exact traveling wave solutions of the system for the Belousov–Zhabotinskii reaction are found. Periodic solutions expressed in terms of the Weierstrass elliptic function are also given.

Explicit Solution to the Constant Radial Acceleration Problem

Although the constant radial acceleration problem is known to be integrable and has received some recent attention in an orbital mechanics context, a closed form explicit solution relating the state variables to a time parameter has eluded all researchers so far. It is here shown how such a solution exists and is elegantly expressed in terms of the Weierstrass elliptic and related functions. Previously known facts can be derived from the new explicit solution and new insights are revealed.

Stability analysis of solitary wave solutions for the fourth-order nonlinear Boussinesq water wave equation

In the present study, the nonlinear Boussinesq type equation describe the bi-directional propagation of small amplitude long capillary–gravity waves on the surface of shallow water. By using the extended auxiliary equation method, we obtained some new soliton like solutions for the two-dimensional fourth-order nonlinear Boussinesq equation with constant coefficient. These solutions include symmetrical, non-symmetrical kink solutions, solitary pattern solutions, Jacobi and Weierstrass elliptic function solutions and triangular function solutions. The stability analysis for these solutions are discussed.