Approximation of Elliptic Functions by Means of Trigonometric Functions with Applications

In this work, we study the Duffing equation. Analytical solution for undamped and unforced case is provided for any given arbitrary initial conditions. An approximate analytical solution is given for the damped or trigonometrically forced Duffing equation for arbitrary initial conditions. The analytical solutions are expressed in terms of elementary trigonometric functions as well as in terms of the Jacobian elliptic functions. Examples are added to illustrate the obtained results. We also introduce new functions for approximating the Jacobian and Weierstrass elliptic functions in terms of the trigonometric functions sine and cosine. Results are high accurate.

The field of elliptic functions, apart from its own mathematical beauty, has many applications in physics in a variety of topics, such as string theory or integrable systems. This book, which focuses on the Weierstrass theory of elliptic functions, aims at senior undergraduate and junior graduate students in physics or applied mathematics. Supplemented by problems and solutions, it provides a fast, but thorough introduction to the mathematical theory and presents some important applications in classical and quantum mechanics. Elementary applications, such as the simple pendulum, help the readers develop physical intuition on the behavior of the Weierstrass elliptic and related functions, whereas more Interesting and advanced examples, like the n=1 Lame problem-a periodic potential with an exactly solvable band structure, are also presented.

Non-linear second-order differential equations whose solutions are the elliptic functions sn(t, k), cn(t, k) and dn(t, k) are investigated. Using Mathematica, high precision numerical solutions are generated. From these data, Fourier coefficients are determined yielding approximate formulas for these non-elementary functions that are correct to at least 11 decimal places. These formulas have the advantage over numerically generated data that they are computationally efficient over the entire real line. This approach is seen as further justification for the early introduction of Fourier series in the undergraduate curriculum, for by doing so, models previously considered hard or advanced, whose solution involves elliptic functions, can be solved and plotted as easily as those models whose solutions involve merely trigonometric or other elementary functions.

5 - Indefinite Integrals of Special Functions

Based on the computerized symbolic computation, a new rational expansion method using the Jacobian elliptic function was presented by means of a new general ansätz and the relations among the Jacobian elliptic functions. The results demonstrated an effective direction in terms of a uniformed construction of the new exact periodic solutions for nonlinear differential–difference equations, where two representative examples were chosen to illustrate the applications. Various periodic wave solutions, including Jacobian elliptic sine function, Jacobian elliptic cosine function and the third elliptic function solutions, were obtained. Furthermore, the solitonic solutions and trigonometric function solutions were also obtained within the limit conditions in this paper.

In this paper, we investigate the variable coefficients Sasa-Satsuma model, which can describe the propagation of a light pulse in a cylindrical fiber. We study this model and obtain rich solutions using two separate methods. We obtain analytical Weierstrass elliptic function solutions using the Weierstrass elliptic function expansion method. Some Jacobi elliptic function solutions are obtained using the modified Jacobi elliptic function expansion method. When the Jacobi elliptic function degenerates, we obtain the corresponding trigonometric, hyperbolic function solutions and periodic solutions. We also try to take the coefficients of the equation as some functions and obtain some more complicated exact solutions, which have not appeared in previous studies. Finally, we simulate some waveform diagrams of the solutions using the computer software Mathematica and obtain periodic waves, bright and dark soliton, double solitons and some complex periodic waves. With these waveform diagrams, we can observe the dynamical behavior of the solutions more clearly.

Optical solitons in birefringent fibers with four-wave mixing for quadratic–cubic nonlinearity by F-expansion

Nonlinear oscillations caused by the initial deviation of the oscillator from the equilibrium position or the initial velocity given to it in this position are considered. It is assumed that the restoring force is proportional to the sine of the displacement of the oscillatory system. There are two variants of the sine: trigonometric and hyperbolic. In the first variant, the power characteristic of the oscillator is soft, and in the second, it is rigid. With a soft power characteristic, restrictions on the initial perturbations of the system are introduced. The exact analytic solutions of the nonlinear Cauchy problem are constructed in elliptic functions. Closed formulas for calculating the displacements of the oscillator and the period of cyclic motion are derived and tested by calculations. To simplify the calculations, in the absence of tables of elliptic Jacobi functions, approximate representations of them in elementary functions are proposed. Examples of calculations are given.

The construction of exact solution for higher‐dimensional nonlinear equation plays an important role in knowing some facts that are not simply understood through common observations. In our work, (4 + 1)‐dimensional nonlinear Fokas equation, which is an important physical model, is discussed by using the extended F‐expansion method and its variant. And some new exact solutions expressed by Jacobi elliptic function, Weierstrass elliptic function, hyperbolic function, and trigonometric function are obtained. The related results are enriched.

In this manuscript, we first perform a complete Lie symmetry classification for a higher-dimensional shallow water wave equation and then construct the corresponding reduced equations with the obtained Lie symmetries. Moreover, with the extended F-expansion method, we obtain several new nonlinear wave solutions involving differentiable arbitrary functions, expressed by Jacobi elliptic function, Weierstrass elliptic function, hyperbolic function and trigonometric function.

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 Weierstrass elliptic function expansion method and its applications in nonlinear wave equations

A mathematical representation that includes probability in describing a system’s development is called a stochastic process. A random variable or noise-containing function is generally employed to represent a stochastic term in a mathematical model. In this paper, we analyze optical soliton (OS) solutions for the well-known Stochastic Biswas–Milovic equation (SBME) with the parabolic law nonlinearity. The Sub-ODE approach is used for this purpose. A variety of new optical soliton solutions are generated, including hyperbolic function, periodic solitons, rational solitons, Jacobi elliptic function (JEF), Weierstrass Elliptic Function (WEF), positive solitons, bright solitons, kink type solitons and dark solitons. Localized solitons, such as bright (positive peaks) and dark (localized dips) solitons, are explained by hyperbolic functions. Localized algebraic waves are represented by rational solitons, even with periodic and doubly-periodic solitons are depicted by JEF and WEF, respectively. Kink solitons represent a variety of nonlinear phenomena by connecting different asymptotic states. Bose–Einstein condensation, fiber optic sensors, plasma physics, optical communication and other fields belong to applications for these solitons. Additionally, we will plot graphs to visually represent the system’s response. To plot some graphs for SBME, we will first import the necessary libraries into Jupyter as a machine learning tool, including matplotlib, scipy.integrate and numpy.

Proceedings of the London Mathematical SocietyVolume s1-17, Issue 1 p. 355-379 Articles Some Applications of Weierstrass's Elliptic Functions Mr. A. G. Greenhill, Mr. A. G. GreenhillSearch for more papers by this author Mr. A. G. Greenhill, Mr. A. G. GreenhillSearch for more papers by this author First published: November 1885 https://doi.org/10.1112/plms/s1-17.1.355AboutPDF ToolsRequest permissionExport citationAdd to favoritesTrack citation ShareShare Give accessShare full text accessShare full-text accessPlease review our Terms and Conditions of Use and check box below to share full-text version of article.I have read and accept the Wiley Online Library Terms and Conditions of UseShareable LinkUse the link below to share a full-text version of this article with your friends and colleagues. Learn more.Copy URL Share a linkShare onFacebookTwitterLinked InRedditWechat Volumes1-17, Issue1November 1885Pages 355-379 RelatedInformation

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.