Jacobian Elliptic Research Articles

The Approximate Solving Methods for the Cubic Duffing Equation based on the Jacobi Elliptic Functions

Article The Approximate Solving Methods for the Cubic Duffing Equation based on the Jacobi Elliptic Functions was published on December 1, 2009 in the journal International Journal of Nonlinear Sciences and Numerical Simulation (volume 10, issue 11-12).

New Periodic Solution to Jacobi Elliptic Functions of a (2+1)-Dimensional BKP Equation and a Generalized Klein–Gordon Equation

With the help of the homogeneous balance method, the Jacobi elliptic expansion method and the auxiliary equation method, the first elliptic function equation is used to obtain the Jacobi doubly periodic wave solutions of the (2+1)-dimensional B-type Kadomtsev–Petviashvili (BKP) equation and the generalized Klein–Gordon equation. The method is also valid for other (1+1)-dimensional and higher dimensional systems.

The Distribution of the Zeros of Jacobian Elliptic Functions with Respect to the Parameter k

We investigate the distribution of the zeros of the twelve Jacobian elliptic functions sn(z, k), etc. as functions of k for fixed z. We show that the number of zeros inside a disc given by ¦k¦ ≤ r is approximately of order r2 and hence that the mapping k ↦ sn(z, k) is of order 2.

An analytical solution for a coupled partial differential equation

A variational method given by Ritz has been applied to the Coupled Partial Differential Equation to construct an analytical solution. The solution of Partial Differential Equation gives a good description of both linear and non-linear evolution of instabilities generated in waves due to modulation. The spatially periodic trial function is chosen in the form of combination of Jacobian Elliptic Functions with the dependence of its parameters subject to optimization. This equation reduced to stationary form of Non-linear Schrödinger Equation in the static limit.

F43 異なる二つの楕円関数の和を母解とする楕円型平均法(F4 機械力学(非線形振動・同期化現象))

F43 異なる二つの楕円関数の和を母解とする楕円型平均法(F4 機械力学(非線形振動・同期化現象))

Some New Exact Solutions of Jacobian Elliptic Function of Petviashvili Equation

By using the modified mapping method, we find new exact solutions of the Petviashvili equation. The solutions obtained in this paper include Jacobian elliptic function solutions, combined Jacobian elliptic function solutions, soliton solutions, triangular function solutions.

An Elliptic Averaging Method Using Sum of Jacobian Elliptic Cosine and Sine Functions

An improved averaging method is proposed in order to obtain a highly accurate periodic solution composed of only odd order harmonics in a strongly nonlinear dynamical system. In this method, sum of the Jacobian elliptic cosine (cn) and sine (sn) function is incorporated as the generating solution. The proposed method is applicable to relatively general nonlinear systems based on Duffing equation. The stability of the solution is analyzed by obtaining the characteristic multipliers of the variational equation. The numerical results for typical nonlinear oscillators are shown. The effectiveness of the proposed method is verified by comparing the computational results with those obtained by the shooting method.

Some New Solutions of Jacobi Elliptic Functions to mBBMEquation

The modified mapping method is further improved by the expandedexpression of u(ξ) that contains the terms of the first-orderderivative of function f(ξ). Some new exact solutions to themBBM equation are determined by means of the method. We can obtain manynew solutions in terms of the Jacobi elliptic functions of the equation.

SOLVABLE THREE-DIMENSIONAL RATIONAL CHAOTIC MAP DEFINED BY JACOBIAN ELLIPTIC FUNCTIONS

Cryptanalysis needs a lot of pseudo-random numbers. In particular, a sequence of independent and identically distributed (i.i.d.) binary random variables plays an important role in modern digital communication systems. Sufficient conditions have been recently provided for a class of ergodic maps of an interval onto itself: R1 → R1 and its associated binary function to generate a sequence of i.i.d. random variables. In order to get more i.i.d. binary random vectors, Jacobian elliptic Chebyshev rational map, its derivative and second derivative which define a Jacobian elliptic space curve have been introduced. Using duplication formula gives three-dimensional real-valued sequences on the space curve onto itself: R3 → R3. This also defines three projective onto mappings, represented in the form of rational functions of xn, yn, zn. These maps generate a three-dimensional sequence of i.i.d. random vectors.

Extended Jacobi Elliptic Function Rational ExpansionMethod and Its Application to (2+1)-Dimensional StochasticDispersive Long Wave System

In this work, by means of ageneralized method and symbolic computation, we extend the Jacobielliptic function rational expansion method to uniformly constructa series of stochastic wave solutions for stochastic evolutionequations. To illustrate the effectiveness of our method, we takethe (2+1)-dimensional stochastic dispersive long wave system as anexample. We not only have obtained some known solutions, but alsohave constructed some new rational formal stochastic Jacobielliptic function solutions.

H25 ヤコビのdn関数とzeta関数の結合関数を母解とした平均法 : 分数調波振動の解析(H2 機械力学(振動制御))

H25 ヤコビのdn関数とzeta関数の結合関数を母解とした平均法 : 分数調波振動の解析(H2 機械力学(振動制御))

Jacobi Elliptic Function Solutions of Some NonlinearEvolution Equations

In this letter, the modified Jacobi elliptic function expansionmethod is extended to solve M-coupled KdV equation, M-coupledIto equation, vKdV equation, and AKNS equation. Some new Jacobielliptic function solutions are obtained by using Mathematica.When the modulus m→1, those periodic solutionsdegenerate as the corresponding soliton solutions.

Multiple Jacobi Elliptic Function Solutions toIntegrable Higher Order Broer–Kaup Equationin (2+1)-Dimensional Spaces

In this paper, we improve the method for deriving Jacobielliptic function solutions of nonlinear evolution equations givenin Ref. [12] and apply it to the integrable higher-orderBroer–Kaup system in (2+1)-dimensional spaces. Somenew elliptic function solutions are obtained.

Duality for Jacobi Group Orbit Spaces and Elliptic Solutions of the WDVV Equations

From any given Frobenius manifold one may construct a so-called ’dual’ structure which, while not satisfying the full axioms of a Frobenius manifold, shares many of its essential features, such as the existence of a prepotential satisfying the Witten– Dijkgraaf–Verlinde–Verlinde equations of associativity. Jacobi group orbit spaces naturally carry the structure of a Frobenius manifold and hence there exists a dual prepotential. In this paper this dual prepotential is constructed and expressed in terms of the elliptic polylogarithm function of Beilinson and Levin.

New Extended Jacobi Elliptic Function RationalExpansion Method and Its Application

In this paper, an extendedJacobi elliptic function rational expansion method is proposed forconstructing new forms of exact Jacobi elliptic function solutionsto nonlinear partial differential equations by means of making amore general transformation. For illustration, we apply the methodto the (2+1)-dimensional dispersive long wave equation andsuccessfully obtain many new doubly periodic solutions, whichdegenerate as soliton solutions when the modulus m approximates1. The method can also be applied to other nonlinear partialdifferential equations.

New Modified Jacobi Elliptic Function ExpansionMethod and Its Application to (3+1)-Dimensional KPEquation

With the aid of computerized symbolic computation, the newmodified Jacobi elliptic function expansion method forconstructing exact periodic solutions of nonlinear mathematicalphysics equation is presented by a new general ansätz.The proposed method is more powerful than most of the existingmethods. By use of the method, we not only can successfullyrecover the previously known formal solutions but also can constructnew and more general formal solutions for some nonlinear evolutionequations. We choose the (3+1)-dimensionalKadomtsev–Petviashvili equation to illustrate our method. As aresult, twenty families of periodic solutions are obtained. Ofcourse, more solitary wave solutions, shock wave solutions ortriangular function formal solutions can be obtained at their limitcondition.

Jacobian Elliptic Function Method and Solitary Wave Solutionsfor Hybrid Lattice Equation

In this paper, we have successfully extended the Jacobian ellipticfunction expansion approach to nonlinear differential-differenceequations. The Hybrid lattice equation is chosen to illustratethis approach. As a consequence, twelve families of Jacobian ellipticfunction solutions with different parameters of the Hybrid latticeequation are obtained. When the modulus m→1 or 0,doubly-periodic solutions degenerate to solitonic solutions andtrigonometric function solutions, respectively.

Further Extended Jacobi Elliptic Function Rational Expansion Method andNew Families of Jacobi Elliptic Function Solutions to (2+1)-Dimensional DispersiveLong Wave Equation

In this paper, a further extended Jacobi elliptic function rational expansionmethod is proposed for constructing new forms of exact solutionsto nonlinear partial differential equations by making a moregeneral transformation. For illustration, we apply the method to(2+1)-dimensional dispersive long wave equation andsuccessfully obtain many new doubly periodic solutions. When themodulus m→1, these solutions degenerate as solitonsolutions. The method can be also applied to other nonlinearpartial differential equations.

Exact Jacobian Elliptic Function Solutions to sinh-GordonEquation

In this paper, two transformations are introduced to solvesinh-Gordon equation by using the knowledge of elliptic equationand Jacobian elliptic functions. It is shown that differenttransformations are required in order to obtain more kinds ofsolutions to the sinh-Gordon equation.

150 楕円型平均法に関する研究 : dn関数とzeta関数を結合した母解の場合

150 楕円型平均法に関する研究 : dn関数とzeta関数を結合した母解の場合