A fundamental dichotomy for Julia sets of a family of elliptic functions

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.

The purpose of this chapter is to provide examples of elliptic functions with prescribed properties of the orbits of critical points (and values). We are primarily focused on constructing examples of various classes of compactly nonrecurrent elliptic functions. All these examples are either Weierstrass elliptic functions or their modifications. The dynamics of such functions depends heavily on the lattice. The first three sections of this chapter have a preparatory character and, respectively, describe the basic dynamical and geometric properties of all Weierstrass elliptic functions generated by square and triangular lattices. We then provide simple constructions of many classes of elliptic functions discerned in the previous chapter. We essentially cover all of them. All these examples stem from Weierstrass $\wp$ functions. Finally, we also provide some different, interesting on their own, and historically first examples of various kinds of Weierstrass $\wp$ elliptic functions and their modifications coming from a series of papers by Hawkins and her collaborators.

In this work, we give approximate expressions for Jacobian and elliptic Weierstrass functions and their inverses by means of the elementary trigonometric functions, sine and cosine. Results are reasonably accurate. We show the way the obtained results may be applied to solve nonlinear ODEs and other problems arising in nonlinear physics. The importance of the results in this work consists on giving easy and accurate way to evaluate the main elliptic functions cn, sn, and dn, as well as the Weierstrass elliptic function and their inverses. A general principle for solving some nonlinear problems through elementary functions is stated. No similar approach has been found in the existing literature.

5 - Indefinite Integrals of Special Functions

In this paper, we investigate elliptic functions of the form [Formula: see text], where [Formula: see text] is the Weierstrass elliptic function on a real rhombic lattice. We show that a typical function in this family has a superattracting fixed point at the origin and five other equivalence classes of critical points. We investigate conditions on the lattice which guarantee that [Formula: see text] has a double toral band, and we show that this family contains the first known examples of elliptic functions for which the Julia set is disconnected but not Cantor.

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

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

In the first chapter, we used several times the fact that the derivative of an elliptic function is also an elliptic function with the same periods. However, the opposite statement is not correct; the indefinite integral of an elliptic function is not necessarily an elliptic function. This class of non-elliptic functions typically possess other interesting quasi-periodicity properties. In this chapter we study two such quasi-periodic functions, the zeta and sigma functions, which are derived from the Weierstrass elliptic function. Then, we study their basic properties and express some very useful theorems. Emphasis is given to the theorems that allow the expression of any elliptic function in terms of the aforementioned quasi-periodic functions.

Weierstrass semi-rational expansion method and new doubly periodic solutions of the generalized Hirota–Satsuma coupled KdV system

The Jacobi and Weierstrass elliptic functions used to be part of the standard mathematical arsenal of physics students. They appear as solutions of many important problems in classical mechanics: the motion of a planar pendulum (Jacobi), the motion of a force-free asymmetric top (Jacobi), the motion of a spherical pendulum (Weierstrass) and the motion of a heavy symmetric top with one fixed point (Weierstrass). The planar pendulum can, in fact, be used to highlight an important connection between the Jacobi and Weierstrass elliptic functions. The easy access to mathematical software by physics students suggests that they might reappear as useful mathematical tools in the undergraduate curriculum.

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.

A solution for the new Hamiltonian amplitude equation is derived using a property of the reciprocal Weierstrass elliptic function. The Weierstrass elliptic function solution has been subsequently expressed in terms of the Jacobian elliptic function. The solitary wave solution which is the infinite period counterpart of the Jacobian elliptic function solution has also been derived.

In numerous fields of mathematical physics, including nuclear physics, fluid dynamics, quantum optics, and plasma physics, the idea of nonlinear evolution equation has left an enduring impression. The concept of concatenation model has recently gained its popularity after its first appearance during 2014. Such a model was proposed in nonlinear optics and exists in two forms: the concatenation model and the dispersive concatenation model, both of which depend on the fundamental components concatenated for their formulation. Likewise, the current paper proposes a concatenation model from plasma physics whose fundamental components are the Kaup–Newell equation, Chen–Lee–Liu equation and the Gerdjikov–Ivanov equation. These encompass various concepts such as Langmuir waves, Alfvén waves, and cold plasmas, which are commonly studied in plasma physics. The special cases of this newly structured concatenation model are apparent as discussed in detail in the subsequent section.Next, the model’s integrability is examined. For this recently developed model, the soliton solutions are obtained using two integration techniques. The methods are the enhanced direct algebraic method and the modified sub-ODE (ordinary differential equation) approach. These two approaches use the intermediary Jacobi’s and Weierstrass’ elliptic functions, respectively, to obtain the soliton solutions. Solitons can be used to identify special cases of these functions. We utilize the parameter constraints that naturally arise from the two integration approaches. Following an introduction to the model, these are covered in detail in the remaining text.The concatenated DNLS model formulated here offers a single parameterized framework in which KN, CLL and GI arise as embedded limits; the tunable derivative couplings and higher-order amplitudinal terms enable controlled passage between convective self-steepening, mixed derivative nonlinearities and quintic saturation. This structure is novel in providing a unified description that consolidates previously separate DNLS-type models into a single tractable form, thereby enabling systematic exploration of plasma nonlinearities across distinct physical regimes.•The paper proposes a novel concatenation model in plasma physics constructed from three fundamental equations—the Kaup–Newell, Chen–Lee–Liu, and Gerdjikov–Ivanov equations—representing key plasma wave phenomena such as Langmuir and Alfvén waves.•Soliton solutions of the model are analytically derived using two powerful integration techniques: the enhanced direct algebraic method (involving Jacobi elliptic functions) and the modified sub-ODE method (utilizing Weierstrass elliptic functions).•The study identifies integrability conditions and parameter constraints from both solution approaches, offering insight into special cases of the model and contributing to the theoretical understanding of nonlinear plasma wave dynamics.

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