scholarly journals Jacobi Elliptic Solutions for Nonlinear Differential Difference Equations in Mathematical Physics

2012 ◽  
Vol 2012 ◽  
pp. 1-15 ◽  
Author(s):  
Khaled A. Gepreel ◽  
A. R. Shehata
Keyword(s):  
Exact Solutions ◽  
Difference Equations ◽  
Elliptic Functions ◽  
Mathematical Physics ◽  
Jacobi Elliptic Function ◽  
Lattice Equation ◽  
Discrete Nonlinear Schrödinger Equation ◽  
Saturable Nonlinearity ◽  
New Exact Solutions ◽  
Elliptic Solutions

We put a direct new method to construct the rational Jacobi elliptic solutions for nonlinear differential difference equations which may be called the rational Jacobi elliptic functions method. We use the rational Jacobi elliptic function method to construct many new exact solutions for some nonlinear differential difference equations in mathematical physics via the lattice equation and the discrete nonlinear Schrodinger equation with a saturable nonlinearity. The proposed method is more effective and powerful to obtain the exact solutions for nonlinear differential difference equations.

scholarly journals Exact Solutions for Nonlinear Differential Difference Equations in Mathematical Physics

2013 ◽  
Vol 2013 ◽  
pp. 1-10 ◽  
Author(s):  
Khaled A. Gepreel ◽  
Taher A. Nofal ◽  
Fawziah M. Alotaibi
Keyword(s):  
Solitary Wave ◽  
Exact Solutions ◽  
Difference Equations ◽  
Toda Lattice ◽  
Expansion Method ◽  
Mathematical Physics ◽  
Discrete Nonlinear Schrödinger Equation ◽  
Nonlinear Schrödinger ◽  
Saturable Nonlinearity ◽  
Relativistic Toda Lattice

We modified the truncated expansion method to construct the exact solutions for some nonlinear differential difference equations in mathematical physics via the general lattice equation, the discrete nonlinear Schrodinger with a saturable nonlinearity, the quintic discrete nonlinear Schrodinger equation, and the relativistic Toda lattice system. Also, we put a rational solitary wave function method to find the rational solitary wave solutions for some nonlinear differential difference equations. The proposed methods are more effective and powerful to obtain the exact solutions for nonlinear difference differential equations.

scholarly journals The Modified Rational Jacobi Elliptic Functions Method for Nonlinear Differential Difference Equations

2012 ◽  
Vol 2012 ◽  
pp. 1-30 ◽  
Author(s):  
Khaled A. Gepreel ◽  
Taher A. Nofal ◽  
Ali A. Al-Thobaiti
Keyword(s):  
Schrödinger Equation ◽  
Exact Solutions ◽  
Nonlinear Schrödinger Equation ◽  
Difference Equations ◽  
Schrodinger Equation ◽  
Elliptic Functions ◽  
Mathematical Physics ◽  
Nonlinear Schrodinger Equation ◽  
Jacobi Elliptic Functions ◽  
Discrete Nonlinear Schrödinger Equation

We modified the rational Jacobi elliptic functions method to construct some new exact solutions for nonlinear differential difference equations in mathematical physics via the lattice equation, the discrete nonlinear Schrodinger equation with a saturable nonlinearity, the discrete nonlinear Klein-Gordon equation, and the quintic discrete nonlinear Schrodinger equation. Some new types of the Jacobi elliptic solutions are obtained for some nonlinear differential difference equations in mathematical physics. The proposed method is more effective and powerful to obtain the exact solutions for nonlinear differential difference equations.

scholarly journals New Jacobi Elliptic Function Solutions for the Kudryashov-Sinelshchikov Equation Using Improved F-Expansion Method

2013 ◽  
Vol 2013 ◽  
pp. 1-6 ◽  
Author(s):  
Yinghui He
Keyword(s):  
Exact Solutions ◽  
Elliptic Function ◽  
Evolution Equations ◽  
Nonlinear Evolution ◽  
Expansion Method ◽  
Mathematical Physics ◽  
Nonlinear Evolution Equations ◽  
Hyperbolic Function ◽  
Jacobi Elliptic Function ◽  
New Exact Solutions

Based on the F-expansion method with a new subequation, an improved F-expansion method is introduced. As illustrative examples, some new exact solutions expressed by the Jacobi elliptic function of the Kudryashov-Sinelshchikov equation are obtained. When the modulusmof the Jacobi elliptic function is driven to the limits 1 and 0, some exact solutions expressed by hyperbolic function and trigonometric function can also be obtained. The method is straightforward and concise and is promising and powerful for other nonlinear evolution equations in mathematical physics.

scholarly journals New Jacobi Elliptic Function Solutions for the Zakharov Equations

2012 ◽  
Vol 2012 ◽  
pp. 1-16 ◽  
Author(s):  
Yun-Mei Zhao
Keyword(s):  
Exact Solutions ◽  
Elliptic Function ◽  
Evolution Equations ◽  
Nonlinear Evolution ◽  
Expansion Method ◽  
Mathematical Physics ◽  
Nonlinear Evolution Equations ◽  
Jacobi Elliptic Function

A generalized(G′/G)-expansion method is proposed to seek the exact solutions of nonlinear evolution equations. Being concise and straightforward, this method is applied to the Zakharov equations. As a result, some new Jacobi elliptic function solutions of the Zakharov equations are obtained. This method can also be applied to other nonlinear evolution equations in mathematical physics.

Explicit solutions to nonlinear Chen–Lee–Liu equation

2021 ◽  
pp. 2150438
Author(s):  
Lanre Akinyemi ◽  
Najib Ullah ◽  
Yasir Akbar ◽  
Mir Sajjad Hashemi ◽  
Arzu Akbulut ◽  
...  
Keyword(s):  
Exact Solutions ◽  
Nonlinear Partial Differential Equations ◽  
Elliptic Functions ◽  
Expansion Method ◽  
Nonlinear Phenomena ◽  
Explicit Solutions ◽  
Mathematical Algorithm ◽  
New Exact Solutions ◽  
Order Polynomial ◽  
Fourth Order Polynomial

In this work, a generalized [Formula: see text]-expansion method has been used for solving the nonlinear Chen–Lee–Liu equation. This method is a more common, general, and powerful mathematical algorithm for finding the exact solutions of nonlinear partial differential equations (NPDEs), where [Formula: see text] follows the Jacobi elliptic equation [Formula: see text], and we let [Formula: see text] be a fourth-order polynomial. Many new exact solutions such as the hyperbolic, rational, and trigonometric solutions with different parameters in terms of the Jacobi elliptic functions are obtained. The distinct solutions obtained in this paper clearly explain the importance of some physical structures in the field of nonlinear phenomena. Also, this method deals very well with higher-order nonlinear equations in the field of science. The numerical results described in the plots were obtained by using Maple.

Painlevé Properties And Exact Solutions Of The Generalized Coupled Kdv Equations

2005 ◽  
Vol 60 (5) ◽  
pp. 313-320 ◽  
Author(s):  
Li-Jun Ye ◽  
Ji Lin
Keyword(s):  
Exact Solutions ◽  
Elliptic Function ◽  
Jacobi Elliptic Function ◽  
Negative Index ◽  
Kp Hierarchy ◽  
New Exact Solutions ◽  
Kdv Equations ◽  
Function Expansion ◽  
Coupled Kdv Equations ◽  
Jacobi Elliptic Function Expansion

The generalized coupled Korteweg-de Vries (GCKdV) equations as one case of the four-reduction of the Kadomtsev-Petviashvili (KP) hierarchy are studied in details. The Painlevé properties of the model are proved by using the standard Weiss-Tabor-Carnevale (WTC) method, invariant, and perturbative Painlev´e approaches. The meaning of the negative index k = −2 is shown, which is indistinguishable from the index k = −1. Using the standard and nonstandard Painlevé truncation methods and the Jacobi elliptic function expansion approach, some types of new exact solutions are obtained.

Exact Solutions for Fractional Differential-Difference Equations by (G'/G)-Expansion Method with Modified Riemann-Liouville Derivative

2016 ◽  
Vol 8 (2) ◽  
pp. 293-305 ◽  
Author(s):  
Ahmet Bekir ◽  
Ozkan Guner ◽  
Burcu Ayhan ◽  
Adem C. Cevikel
Keyword(s):  
Difference Equation ◽  
Exact Solutions ◽  
Difference Equations ◽  
Applied Mathematics ◽  
Expansion Method ◽  
New Exact Solutions ◽  
Liouville Derivative ◽  
Nonlinear Lattice ◽  
Fractional Differential ◽  
Differential Difference Equation

AbstractIn this paper, the (G'/G)-expansion method is suggested to establish new exact solutions for fractional differential-difference equations in the sense of modified Riemann-Liouville derivative. The fractional complex transform is proposed to convert a fractional partial differential difference equation into its differential difference equation of integer order. With the aid of symbolic computation, we choose nonlinear lattice equations to illustrate the validity and advantages of the algorithm. It is shown that the proposed algorithm is effective and can be used for many other nonlinear lattice equations in mathematical physics and applied mathematics.

scholarly journals New Exact Solutions for a Higher-Order Wave Equation of KdV Type Using Extended F-Expansion Method

2013 ◽  
Vol 2013 ◽  
pp. 1-8 ◽  
Author(s):  
Yinghui He ◽  
Yun-Mei Zhao ◽  
Yao Long
Keyword(s):  
Exact Solutions ◽  
Wave Equations ◽  
Traveling Wave Solutions ◽  
Type Equation ◽  
Wave Model ◽  
Expansion Method ◽  
Trigonometric Function ◽  
Hyperbolic Function ◽  
Jacobi Elliptic Function ◽  
New Exact Solutions

The F-expansion method is used to find traveling wave solutions to various wave equations. By giving more solutions of the general subequation, an extended F-expansion method is introduced by Emmanuel. In our work, a generalized KdV type equation of neglecting the highest-order infinitesimal term, which is an important water wave model, is discussed by using the extended F-expansion method. And when the parameters satisfy certain relations, some new exact solutions expressed by Jacobi elliptic function, hyperbolic function, and trigonometric function are obtained. The related results are enriched.

scholarly journals AKNS Formalism and Exact Solutions of KdV and Modified KdV Equations with Variable-Coefficients

2016 ◽  
Vol 6 ◽  
pp. 32-41 ◽  
Author(s):  
Supratim Das ◽  
Dibyendu Ghosh
Keyword(s):  
Exact Solutions ◽  
Kdv Equation ◽  
Elliptic Functions ◽  
Variable Coefficients ◽  
Jacobian Elliptic Functions ◽  
Generalized Kdv Equation ◽  
New Exact Solutions ◽  
Kdv Equations ◽  
Modified Kdv Equation ◽  
Modified Kdv Equations

We apply the AKNS hierarchy to derive the generalized KdV equation andthe generalized modified KdV equation with variable-coefficients. We system-atically derive new exact solutions for them. The solutions turn out to beexpressible in terms of doubly-periodic Jacobian elliptic functions.

scholarly journals New Exact Superposition Solutions to KdV2 Equation

2018 ◽  
Vol 2018 ◽  
pp. 1-9 ◽  
Author(s):  
Piotr Rozmej ◽  
Anna Karczewska
Keyword(s):  
Exact Solutions ◽  
Water Waves ◽  
Kdv Equation ◽  
Order Approximation ◽  
Elliptic Functions ◽  
Periodic Functions ◽  
Shallow Water Waves ◽  
First Order ◽  
New Exact Solutions ◽  
First Order Approximation

New exact solutions to the KdV2 equation (also known as the extended KdV equation) are constructed. The KdV2 equation is a second-order approximation of the set of Boussinesq’s equations for shallow water waves which in first-order approximation yields KdV. The exact solutionsA/2dn2[B(x-vt),m]±m cn[B(x-vt),m]dn[B(x-vt),m]+Din the form of periodic functions found in the paper complement other forms of exact solutions to KdV2 obtained earlier, that is, the solitonic ones and periodic ones given by singlecn2ordn2Jacobi elliptic functions.

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