Elliptic Modulus Research Articles

Optical solitons in birefringent fibers with Sasa–Satsuma equation having multiplicative noise with Itô calculus

This paper obtains soliton solutions to Sasa–Satsuma equation, which is studied in birefringent fibers with multiplicative noise. The extended auxiliary equation approach reveals solutions in terms of Jacobi’s elliptic functions from which bright, dark and singular solitons emerge when the limiting operation is applied to the modulus of ellipticity, with the limit approaching zero or unity.

Stationary optical solitons with complex Ginzburg-Landau equation having nonlinear chromatic dispersion.

The current work is on the retrieval of stationary soliton solutions to the complex Ginzburg–Landau equation that is studied with nonlinear chromatic dispersion having a plethora of nonlinear refractive index structures. The Jacobi’s elliptic function approach is employed to recover doubly periodic waves which leads to soliton solutions when the limiting value of the modulus of ellipticity is reached.

Novel Analytical and Numerical Approximations to the Forced Damped Parametric Driven Pendulum Oscillator: Chebyshev Collocation Method

In this work, some novel approximate analytical and numerical solutions to the forced damped driven nonlinear (FDDN) pendulum equation and some relation equations of motion on the pivot vertically for arbitrary angles are obtained. The analytical approximation is derived in terms of the Jacobi elliptic functions with arbitrary elliptic modulus. For the numerical approximations, the Chebyshev collocation numerical method is introduced for analyzing the equation of motion. Moreover, the analytical approximation and numerical approximation using the Chebyshev collocation numerical method and the MATHEMATICA command Fit are compared with the Runge–Kutta (RK) numerical solution. Also, the maximum distance error to all obtained approximations is estimated with respect to the RK numerical solution. The obtained results help many authors to understand the mechanism of many phenomena related to the plasma physics, classical mechanics, quantum mechanics, optical fiber, and electronic circuits.

Quasi-Hopf twist and elliptic Nekrasov factor

We investigate the quasi-Hopf twist of the quantum toroidal algebra of {mathfrak{gl}}_1 as an elliptic deformation. Under the quasi-Hopf twist the underlying algebra remains the same, but the coproduct is deformed, where the twist parameter p is identified as the elliptic modulus. Computing the quasi-Hopf twist of the R matrix, we uncover the relation to the elliptic lift of the Nekrasov factor for instanton counting of the quiver gauge theories on ℝ4× T2. The same R matrix also appears in the commutation relation of the intertwiners, which implies an elliptic quantum KZ equation for the trace of intertwiners. We also show that it allows a solution which is factorized into the elliptic Nekrasov factors and the triple elliptic gamma function.

Formation of rogue waves on the periodic background in a fifth-order nonlinear Schrödinger equation

We construct rogue wave solutions of a fifth-order nonlinear Schrödinger equation on the Jacobian elliptic function background. By combining Darboux transformation and the nonlinearization of spectral problem, we generate rogue wave solution on two different periodic wave backgrounds. We analyze the obtained solutions for different values of system parameter and point out certain novel features of our results. We also compute instability growth rate of both dn and cn periodic background waves for the considered system through spectral stability problem. We show that instability growth rate decreases (increases) for dn-(cn) periodic waves when we vary the value of the elliptic modulus parameter.

Rogue waves on an elliptic function background in complex modified Korteweg–de Vries equation

With the assistance of one fold Darboux transformation formula, we derive rogue wave solutions of the complex modified Korteweg–de Vries equation on an elliptic function background. We employ an algebraic method to find the necessary squared eigenfunctions and eigenvalues. To begin we construct the elliptic function background. Then, on top of this background, we create a rogue wave. We demonstrate the outcome for three distinct elliptic modulus values. We find that when we increase the modulus value the amplitude of rogue waves on the dn-periodic background decreases whereas it increases in the case of cn-periodic background.

Comment on “Solitons in magneto-optics waveguides for the nonlinear Biswas–Milovic equation with Kudryashov’s law of refractive index using the unified auxiliary equation method”

In a recent paper (Elsayed et al., 2021), authors introduce the coupled system of magneto-optics waveguides for the nonlinear Biswas–Milovic equation with Kudryashov’s law of refractive index. Using the unified auxiliary equation method, they construct, in terms of Jacobi elliptic function, mathematical solutions to this system. In some limit cases of the elliptic modulus, they derive bright, dark, singular solitons. Other mathematical solutions are also presented. In this Comment we provide sufficient theoretical evidence showing that the unified auxiliary equation method used in their work leads to wrong solutions.

Exponential Riordan arrays and Jacobi elliptic functions

This paper establishes relationships between elliptic functions and Riordan arrays leading to new classes of Riordan arrays which here are called elliptic Riordan arrays. In particular, the case of Riordan arrays derived from Jacobi elliptic functions that are parameterized by the elliptic modulus k will be treated here. Some concrete examples of such Riordan arrays are presented via a recursive formula.

Closed Analytic Formulas for the Approximation of the Legendre Complete Elliptic Integrals of the First and Second Kinds

The author proposes two sets of closedanalytic functions for the approximate calculus of thecomplete elliptic integrals of the first and secondkinds in the normal form due to Legendre, therespective expressions having a remarkablesimplicity and accuracy. The special usefulness of theproposed formulas consists in that they allowperforming the analytic study of variation of thefunctions in which they appear, by using thederivatives. Comparative tables including theapproximate values obtained by applying the two setsof formulas and the exact values, reproduced fromspecial functions tables are given (all versus therespective elliptic integrals modulus, k = sin ). It is tobe noticed that both sets of approximate formulas aregiven neither by spline nor by regression functions,but by asymptotic expansions, the identity with theexact functions being accomplished for the left end k= 0 ( = 0) of the domain. As one can see, the secondset of functions, although something more intricate,gives more accurate values than the first one andextends itself more closely to the right end k = 1 ( =90) of the domain. For reasons of accuracy, it isrecommended to use the first set until  = 70.5 only,and if it is necessary a better accuracy or a greaterupper limit of the validity domain, to use the secondset, but on no account beyond  = 88.2.

Optical Soliton Solutions to Gerdjikov-Ivanov Equation Without Four-Wave Mixing Terms in Birefringent Fibers by Extended Trial Function Scheme

Without four-wave mixing terms in birefringent fibers, the extended trial function scheme was used to obtain optical soliton solutions for the coupled system corresponding to the Gerdjikov-Ivanov equation. The procedure reveals singular soliton solutions, bright soliton solutions, and highly important solutions in terms of Jacobi’s elliptic function. And in the limiting case of the modulus of ellipticity, singular and singular-periodic soliton solutions, along with their respective existence criteria.

Optical soliton perturbation with Kudryashov’s law of refractive index by modified sub-ODE approach

This paper implements the sub-ODE method and a wide spectrum of solitons are recovered for Kudryashov’s law of refractive index. The self-phase modulation comprises of four nonlinear components of refractive index. The perturbation terms are all of Hamiltonian type and are considered with maximum intensity. The solutions are written in terms of Weierstrass’ elliptic functions and Jacobi’s elliptic function. With the modulus of ellipticity approaching zero or unity, soliton solutions emerge.

Rogue waves for the fourth-order nonlinear Schrödinger equation on the periodic background.

In this paper, we construct rogue wave solutions on the periodic background for the fourth-order nonlinear Schrödinger (NLS) equation. First, we consider two types of the Jacobi elliptic function solutions, i.e., dn- and cn-function solutions. Both dn- and cn-periodic waves are modulationally unstable with respect to the long-wave perturbations. Second, on the background of both periodic waves, we derive rogue wave solutions by combining the method of nonlinearization of spectral problem with the Darboux transformation method. Furthermore, by the study of the dynamics of rogue waves, we find that they have the analogs in the standard NLS equation, and the higher-order effects do not have effect on the magnification factor of rogue waves. In addition, when the elliptic modulus approaches 1, rogue wave solutions can reduce to multi-pole soliton solutions in which the interacting solitons form weakly bound states.

Optical solitons of space-time fractional Sasa-Satsuma equation by F-expansion method

For the space-time fractional Sasa-Satsuma equation, the Atangana's conformable derivative is used to convert the fractional differential equation into the ordinary differential equation. A variety of solutions are obtained by the F-expansion method for the space-time fractional Sasa-Satsuma equation. The exact solutions include Jacobian elliptic function solutions and when the elliptic modulus takes the extreme value, the bright, dark, singular and dark-singular optical soliton solutions are obtained. These solutions contribute to a more accurate interpretation of complex physical problems described by the space-time fractional Sasa-Satsuma equation. After selecting some appropriate parameter values, the three-dimensional and two-dimensional graphs of some obtained solutions are plotted. The effects of the fractional order derivative on the soliton solutions are discussed.

Soliton solutions of Kundu-Eckhaus equation in birefringent optical fiber with inter-modal dispersion

We study the dynamics of optical soliton in birefringent optical fiber with the effect of inter-modal dispersion. We consider the model of the Kundu-Eckhaus (KE) equation for birefringent optical fiber with the inclusion of inter-modal dispersion. Multisoliton solutions of KE equation with the presence of inter-modal dispersion are found analytically by using mapping method and a symbolic computation system. Initially, the solutions are expressed in terms of the Jacobi elliptic function which is also recovered by the use of limiting case of modulus of ellipticity. With the help of graphical illustrations of our solutions, we understand the effect of inter-modal dispersion on the soliton evolution in optical birefringent optical fiber.

Solitions in magneto–optic waveguides with anti–cubic nonlinearity

Optical soliton solutions are recovered for magneto–optic waveguides that maintains anti–cubic form of nonlinear refractive index. The analytical scheme is Jacobi's elliptic function approach. Once the solutions to the governing model are obtained in terms of Jacobi's elliptic functions, the limiting value to it's modulus of ellipticity reveals the complete spectrum of soliton solutions.

Solitons in magneto–optic waveguides with dual–power law nonlinearity

This paper applies new ϕ6 model expansion scheme to obtain solitons in magneto–optic waveguides that come with dual–power law nonlinearity. Initially, the solutions are in terms of Jacobi's elliptic functions. Upon approaching the limiting values of the modulus of ellipticity, bright, dark and singular solitons emerge. The existence criteria of such solitons are also presented.

Analytical solution of the Duffing equation

PurposeThe purpose of this paper is to find an exact analytical expression for the periodic solutions of the double-hump Duffing equation and an expression for the period of these solutions.Design/methodology/approachThe double-hump Duffing equation is presented as a Hamiltonian system and a phase portrait of this system has been found. On the ground of analytical calculations performed using Hamiltonian-based technique, the periodic solutions of this system are represented by Jacobi elliptic functions sn, cn and dn.FindingsExpressions for the periodic solutions and their periods of the double-hump Duffing equation have been found. An expression for the solution, in the time domain, corresponding to the heteroclinic trajectory has also been found. An important element in various applications is the relationship obtained between constant Hamiltonian levels and the elliptic modulus of the elliptic functions.Originality/valueThe results obtained in this paper represent a generalization and improvement of the existing ones. They can find various applications, such as analysis of limit cycles in perturbed Duffing equation, analysis of damped and forced Duffing equation, analysis of nonlinear resonance and analysis of coupled Duffing equations.

An application of hypergeometric functions to heat kernels on rectangular and hexagonal tori and a \u201cWeltkonstante\u201d-or-how Ramanujan split temperatures

In this work we investigate the heat kernel of the Laplace–Beltrami operator on a rectangular torus and the according temperature distribution. We compute the minimum and the maximum of the temperature on rectangular tori of fixed area by means of Gauss’ hypergeometric function _2F_1 and the elliptic modulus. In order to be able to do this, we employ a beautiful result of Ramanujan, connecting hypergeometric functions, the elliptic modulus and theta functions. Also, we investigate the temperature distribution of the heat kernel on hexagonal tori and use Ramanujan’s corresponding theory of signature 3 to derive analogous results to the rectangular case. Lastly, we show connections to the problem of finding the exact value of Landau’s “Weltkonstante”, a universal constant arising in the theory of extremal holomorphic mappings; and for a related, restricted extremal problem we show that the conjectured solution is the second lemniscate constant.

Optical dromions in cascaded systems with a couple of integration norms

This paper revisits cascaded system with Kerr nonlinearity but in (2 + 1)-dimensions. Two forms of integration architecture lead to bright, dark and singular solitons to the model that will be useful in a variety of optoelectronic devices such as optical couplers, magneto-optic waveguides. Jacobi’s elliptic function leads to soliton solutions after considering limiting values to the modulus of ellipticity.

Optical solitons with complex Ginzburg–Landau equation for two nonlinear forms using F-expansion

This paper implements F-expansion scheme to obtain Jacobi’s elliptic function to complex Ginzburg–Landau equation with two nonlinear forms. In the limiting case of the modulus of ellipticity, bright and dark soliton solutions emerge.