Weierstrass Elliptic Function Research Articles

Elliptic traveling waves of the Olver equation

Nonlinear waves on water are studied. The method recently developed by Demina and Kudryashov is applied to the Olver water wave equation. New solutions of this equation are found. These solutions are expressed in terms of the Weierstrass elliptic function.

Conformal Mapping of Circular Quadrilaterals and Weierstrass Elliptic Functions

Numerical and theoretical aspects of conformal mappings from a disk to a circular-arc quadrilateral, symmetric with respect to the coordinate axes, are developed. The problem of relating the accessory parameters (prevertices together with coefficients in the Schwarzian derivative) to the geometric parameters is solved numerically, including the determination of the parameters for univalence. The study involves the related mapping from an appropriate Euclidean rectangle to the circular-arc quadrilateral. Its Schwarzian derivative involves the Weierstrass ℘-function, and consideration of this related mapping problem leads to some new formulas concerning the zeroes and the images of the half-periods of ℘.

The First‐Integral Method and Abundant Explicit Exact Solutions to the Zakharov Equations

This paper is concerned with the system of Zakharov equations which involves the interactions between Langmuir and ion‐acoustic waves in plasma. Abundant explicit and exact solutions of the system of Zakharov equations are derived uniformly by using the first integral method. These exact solutions are include that of the solitary wave solutions of bell‐type for n and E, the solitary wave solutions of kink‐type for E and bell‐type for n, the singular traveling wave solutions, periodic wave solutions of triangle functions, Jacobi elliptic function doubly periodic solutions, and Weierstrass elliptic function doubly periodic wave solutions. The results obtained confirm that the first integral method is an efficient technique for analytic treatment of a wide variety of nonlinear systems of partial differential equations.

Surfaces immersed in Lie algebras associated with elliptic integrals

The objective of this work is to adapt the Fokas–Gel’fand immersion formula to ordinary differential equations written in the Lax representation. The formalism of generalized vector fields and their prolongation structure is employed to establish necessary and sufficient conditions for the existence and integration of immersion functions for surfaces in Lie algebras. As an example, a class of second-order, integrable, ordinary differential equations is considered and the most general solutions for the wavefunctions of the linear spectral problem are found. Several explicit examples of surfaces associated with Jacobian and -Weierstrass elliptic functions are presented.

The Poisson-Boltzmann theory for the two-plates problem: Some exact results

The general solution to the nonlinear Poisson-Boltzmann equation for two parallel charged plates, either inside a symmetric electrolyte, or inside a 2q:-q asymmetric electrolyte, is found in terms of Weierstrass elliptic functions. From this we derive some exact asymptotic results for the interaction between charged plates, as well as the exact form of the renormalized surface charge density.

Exact solutions for the transformed reduced Ostrovsky equation via the [formula omitted]-expansion method in terms of Weierstrass-elliptic and Jacobian-elliptic functions

Although the F-expansion method is a relatively old method for obtaining exact solutions for nonlinear partial differential equations, its advantage over the other auxiliary equation methods is shown in this work for solving the transformed reduced Ostrovsky equation. In fact, to the authors’ knowledge, this is the first time that a largest number of exact solutions for the auxiliary equation F′(ω)=PF4(ω)+QF2(ω)+R has been presented. It is used in this work to get 52 types of exact solution: six for Weierstrass-elliptic function solutions and the rest for Jacobian-elliptic function solutions. However, in consideration of Parkes (2010) [9], who states that the term ‘new’ has to be used very carefully, it is very important to mention that the solutions obtained are not completely different. Many examples for proving Parkes’s view [9] have been examined. Moreover, soliton-like solutions and trigonometric-function solutions are also obtained as limiting cases. The previous results determined by the hyperbolic tangent method (Yusufoğlu and Bekir, 2007) [5] and exponential function approach (Kangalgil and Ayaz, 2008) [6] are found to be limited and special cases of the current results.

On the integrability of a Hamiltonian reduction of a 2+1-dimensional non-isothermal rotating gas cloud system

A 2+1-dimensional version of a non-isothermal gas dynamic system with origins in the work of Ovsiannikov and Dyson on spinning gas clouds is shown to admit a Hamiltonian reduction which is completely integrable when the adiabatic index γ = 2. This nonlinear dynamical subsystem is obtained via an elliptic vortex ansatz which is intimately related to the construction of a Lax pair in the integrable case. The general solution of the gas dynamic system is derived in terms of Weierstrass (elliptic) functions. The latter derivation makes use of a connection with a stationary nonlinear Schrödinger equation and a Steen–Ermakov–Pinney equation, the superposition principle of which is based on the classical Lamé equation.

Connectivity of Julia sets for Weierstrass elliptic functions on square lattices

We show that the Julia set of a Weierstrass elliptic function on a square lattice is connected. The techniques used to prove this result are used to show a similar result for a related family of rational maps obtained from the Laurent series.

A METHOD TO CONSTRUCT WEIERSTRASS ELLIPTIC FUNCTION SOLUTION FOR NONLINEAR EQUATIONS

A method for constructing the solutions of nonlinear evolution equations by using the Weierstrass elliptic function and its first-order derivative was presented. This technique was then applied to Burgers and Klein–Gordon equations which showed its efficiency and validality for exactly some solving nonlinear evolution equations.

Exact analytical solutions of three-dimensional Gross—Pitaevskii equation with time—space modulation

By the generalized sub-equation expansion method and symbolic computation, this paper investigates the (3 + 1)-dimensional Gross—Pitaevskii equation with time- and space-dependent potential, time-dependent nonlinearity, and gain or loss. As a result, rich exact analytical solutions are obtained, which include bright and dark solitons, Jacobi elliptic function solutions and Weierstrass elliptic function solutions. With computer simulation, the main evolution features of some of these solutions are shown by some figures. Nonlinear dynamics of a soliton pulse is also investigated under the different regimes of soliton management.

Poisson-Boltzmann theory for two parallel uniformly charged plates

We solve the nonlinear Poisson-Boltzmann equation for two parallel and like-charged plates both inside a symmetric electrolyte, and inside a 2:1 asymmetric electrolyte, in terms of Weierstrass elliptic functions. From these solutions we derive the functional relation between the surface charge density, the plate separation, and the pressure between plates. For the one plate problem, we obtain exact expressions for the electrostatic potential and for the renormalized surface charge density, both in symmetric and in asymmetric electrolytes. For the two plate problems, we obtain new exact asymptotic results in various regimes.

New generalized Jacobi elliptic function rational expansion method

In this work, a new generalized Jacobi elliptic function rational expansion method is based upon twenty-four Jacobi elliptic functions and eight double periodic Weierstrass elliptic functions, which solve the elliptic equation ϕ ′ 2 = r + p ϕ 2 + q ϕ 4 , is described. As a consequence abundant new Jacobi–Weierstrass double periodic elliptic functions solutions for (3+1)-dimensional Kadmtsev–Petviashvili (KP) equation are obtained by using this method. We show that the new method can be also used to solve other nonlinear partial differential equations (NPDEs) in mathematical physics.

On Weierstrass elliptic function solutions for a ( N+1) dimensional potential KdV equation

This paper is concerned with several aspects of travelling wave solutions for a ( N+1) dimensional potential KdV equation. The Weierstrass elliptic function solutions, the Jaccobi elliptic function solutions, solitary wave solutions, periodic wave solutions to the equation are acquired under certain circumstances. It is shown that the coefficients of derivative terms in the equation cause the qualitative changes of physical structures of the solutions.

Schwarzschild Geodesics in Terms of Elliptic Functions and the Related Red Shift

Using Weierstrassian elliptic functions the exact geodesics in the Schwarzschild metric are expressed in a simple and most transparent form. The results are useful for analytical and numerical applications. For example we calculate the perihelion precession and the light deflection in the post-Einsteinian approximation. The bounded orbits are computed in the post-Newtonian order. As a topical application we calculate the gravitational red shift for a star moving in the Schwarzschild field.

Travelling waves in microstructure as the exact solutions to the 6th order nonlinear equation

The travelling wave solutions to the nonlinear partial differential equation of 6th order are obtained for a solid having two different spatial scales introduced in the microstructure. The slaving principle method is applied, and the exact explicit solution is found in terms of the doubly periodic Weierstrass elliptic function for the corresponding ODE. Several particular cases are discussed for various parameter values, e.g., the solitary “mexican hat” pulse is found with polarity, depending on microstructure parameters.

New types of exact solutions for nonlinear Schrödinger equation with cubic nonlinearity

Although, many exact solutions were obtained for the cubic Schrödinger equation by many researchers, we obtained in this research not only more exact solutions but also new types of exact solutions in terms of Jacobi-elliptic functions and Weierstrass-elliptic function.

Periodic Wave Solutions of a Generalized KdV-mKdV Equation with Higher-Order Nonlinear Terms

The Jacobin doubly periodic wave solution, the Weierstrass elliptic function solution, the bell-type solitary wave solution, the kink-type solitary wave solution, the algebraic solitary wave solution, and the triangular solution of a generalized Korteweg-de Vries-modified Korteweg-de Vries equation (GKdV-mKdV) with higher-order nonlinear terms are obtained by a generalized subsidiary ordinary differential equation method (Gsub-ODE method for short). The key ideas of the Gsub-ODE method are that the periodic wave solutions of a complicated nonlinear wave equation can be constructed by means of the solutions of some simple and solvable ODE which are called Gsub-ODE with higherorder nonlinear terms

Fan sub-equation method for Wick-type stochastic partial differential equations

An improved algorithm is devised for using Fan sub-equation method to solve Wick-type stochastic partial differential equations. Applying the improved algorithm to the Wick-type generalized stochastic KdV equation, we obtain more general Jacobi and Weierstrass elliptic function solutions, hyperbolic and trigonometric function solutions, exponential function solutions and rational solutions.

Analytic treatment of complete and incomplete geodesics in Taub-NUT space-times

We present the complete set of analytical solutions of the geodesic equation in Taub-NUT space-times in terms of the Weierstrass elliptic functions. We systematically study the underlying polynomials and characterize the motion of test particles by its zeros. Since the presence of the ``Misner string'' in the Taub-NUT metric has led to different interpretations, we consider these in terms of the geodesics of the space-time. In particular, we address the geodesic incompleteness at the horizons discussed by Misner and Taub [C. W. Misner and A. H. Taub, Sov. Phys. JETP 28, 122 (1969) [C. W. MisnerA. H. TaubZh. Eksp. Teor. Fiz. 55, 233 (1968)]], and the analytic extension of Miller, Kruskal and Godfrey [J. G. Miller, M. D. Kruskal, and B. Godfrey, Phys. Rev. D 4, 2945 (1971)], and compare with the Reissner-Nordstr\"om space-time.

Third-order superintegrable systems separating in polar coordinates

A complete classification of quantum and classical superintegrable systems in E2 is presented that allow the separation of variables in polar coordinates and admit an additional integral of motion of order 3 in the momentum. New quantum superintegrable systems are discovered for which the potential is expressed in terms of the sixth Painlevé transcendent or in terms of the Weierstrass elliptic function.