Weierstrass Elliptic Function Research Articles

Cnoidal wave, snoidal wave, and soliton solutions of the D(m,n) equation

This paper studies the D(m, n) equation, which is the generalized version of the Drinfeld-Sokolov equation. The traveling wave hypothesis and exp-function method are applied to integrate this equation. The mapping method and the Weierstrass elliptic function method also display an additional set of solutions. The kink, soliton, shock waves, singular soliton solution, cnoidal and snoidal wave solutions are all obtained by these varieties of integration tools. Mathematics Subject Classification 37K10 · 35Q51 · 35Q55

New exact solutions for the generalized variable-coefficient Gardner equation with forcing term

In this work, we study the generalized variable-coefficient Gardner equation (GVG) with nonlinear terms of any order and forcing term, abundant new exact solutions for the equation are obtained by using the general mapping deformation method with the aid of symbolic computation, which include several kinds of types such as Jacobi elliptic wave-like solutions, soliton-like solutions, trigonometric function solutions, Weierstrass elliptic function solution and rational type solution. Some of them are found for the first time, which shows that the applied method is more powerful and will be used in further works to establish more entirely new exact solutions for other kinds of nonlinear partial differential equations arising in mathematical physics.

A direct method for constructing the traveling wave solutions of a modified generalized Vakhnenko equation

Proposing a direct method, namely, the traveling wave reduction combined with a smart variable transformation, a series of traveling wave solutions are constructed for a modified generalized Vakhnenko equation. These solutions include the Jacobi elliptic function solution, the Weierstrass elliptic function solution and the trigonometric function solution.

Solitons and other nonlinear waves of the Boussinesq equation

This paper studies the dynamics of shallow water waves that are governed by the Boussinesq equations. A few perturbation terms are taken into account. The ansatz method is used to carry out the perturbed Boussinesq equation. Later on, the mapping method is used to extract a few more analytical solutions. Additionally, the Weierstrass elliptic function method is also used to obtain solitary waves and singular soliton solutions. Finally, the Lie symmetry approach is used to extract a few more additional solutions.

A note on “The tanh–coth method combined with the Riccati equation for solving nonlinear coupled equation in mathematical physics”

The recent paper “The tanh–coth method combined with the Riccati equation for solving nonlinear coupled equation in mathematical physics” (J. King Saud Univ. Sci. 23 (2011) 127–132) is analyzed. We show that the authors of this paper solved equations with the well known solutions. One of these equations is the famous Riccati equation and the other equation is one for the Weierstrass elliptic function. We present the general solutions of these equations. As this takes place, 19 solutions by authors do not satisfy the equation but the other 29 solutions can be obtained from the general solutions.

All solutions of the n = 5 Lane–Emden equation

All real solutions of the Lane–Emden equation for n = 5 are obtained in terms of Jacobian and Weierstrass elliptic functions. A new family of solutions is found. It is expressed by remarkably simple formulae involving Jacobian elliptic functions only. The general properties and discrete scaling symmetries of these new solutions are discussed. We also comment on their possible applications.

On quintic Eisenstein series and points of order five of the Weierstrass elliptic functions

We employ a new constructive approach to study modular forms of level five by evaluating the Weierstrass elliptic functions at points of order five on the period parallelogram. A significant tool in our analysis is a nonlinear system of coupled differential equations analogous to Ramanujan’s differential system for the Eisenstein series on SL(2,ℤ). The resulting relations of level five may be written as a coupled system of differential equations for quintic Eisenstein series. Some interesting combinatorial and analytic consequences result, including an alternative proof of a famous identity of Ramanujan involving the Rogers–Ramanujan continued fraction.

Exact solutions of four generalized Benjamin–Bona–Mahony equations with any order

For the four generalized Benjamin–Bona–Mahony equations that occurred in [4,5], we apply the theory of Weierstrass elliptic function to obtain all the solutions in [4,5]. The traveling wave solutions are expressed by the hyperbolic functions and trigonometric functions. The method is direct and concise, and free of the symbolic calculation. Moreover, it is more effective than the method in [5] and is suitable to other equations with any order.

The application of Weierstrass elliptic functions to Schwarzschild null geodesics

In this paper, we focus on analytical calculations involving null geodesics in some spherically symmetric spacetimes. We use Weierstrass elliptic functions to fully describe null geodesics in Schwarzschild spacetime and to derive analytical formulae connecting the values of radial distance at different points along the geodesic. We then study the properties of light triangles in Schwarzschild spacetime and give the expansion of the deflection angle to the second order in both M/r0 and M/b where M is the mass of the black hole, r0 the distance of the closest approach of the light ray and b the impact parameter. We also use the Weierstrass function formalism to analyze other more exotic cases such as Reissner–Nordstrøm null geodesics and Schwarzschild null geodesics in four and six spatial dimensions. Finally we apply Weierstrass functions to describe the null geodesics in the Ellis wormhole spacetime and give an analytic expansion of the deflection angle in M/b.

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.