scholarly journals An application of the new function method to the Zhiber–Shabat equation

2017 ◽  
Vol 7 (3) ◽  
pp. 271-274
Author(s):  
Tolga Aktürk ◽  
Yusuf Gürefe ◽  
Yusuf Pandır
Keyword(s):  
Traveling Wave ◽  
Function Method ◽  
Elliptic Integral ◽  
Traveling Wave Solutions ◽  
Elliptic Functions ◽  
Integral Function ◽  
Jacobi Elliptic Functions ◽  
New Approach ◽  
Function Solution ◽  
Wave Solutions

This paper applies a new approach including the trial equation based on the exponential function in order to find new traveling wave solutions to Zhiber-Shabat equation. By the using of this method, we obtain a new elliptic integral function solution. Also, this solution can be converted into Jacobi elliptic functions solution by a simple transformation.

scholarly journals New function method to the (n+1)-dimensional nonlinear problems

2017 ◽  
Vol 7 (3) ◽  
pp. 234-239 ◽  
Author(s):  
Tolga Aktürk ◽  
Yusuf Gürefe ◽  
Hasan Bulut
Keyword(s):  
Traveling Wave ◽  
Function Method ◽  
Elliptic Integral ◽  
Traveling Wave Solutions ◽  
Nonlinear Problems ◽  
Integral Function ◽  
Jacobi Elliptic Function ◽  
Mathematical Tool ◽  
New Approach ◽  
Physical Problems

In this study, a new approach that assumes  and  is applied to construct the traveling wave solutions of the (N + 1)-dimensional double sine-Gordon and (N + 1)-dimensional double sinh-cosh-Gordon equations. Some new elliptic integral function solutions are respectively obtained by this method, and then these solutions are converted into the Jacobi elliptic function solutions. According these results, one can easily see that this method is very effective mathematical tool for the (N+1)-dimensional nonlinear physical problems.

New soliton solutions of the CH–DP equation using Lie symmetry method

2018 ◽  
Vol 32 (20) ◽  
pp. 1850234 ◽  
Author(s):  
A. H. Abdel Kader ◽  
M. S. Abdel Latif
Keyword(s):  
Traveling Wave ◽  
Soliton Solutions ◽  
Traveling Wave Solutions ◽  
Elliptic Functions ◽  
Dark Soliton ◽  
Lie Symmetry ◽  
Jacobi Elliptic Functions ◽  
Lie Symmetry Method ◽  
Wave Solutions ◽  
Symmetry Method

In this paper, using Lie symmetry method, we obtain some new exact traveling wave solutions of the Camassa–Holm–Degasperis–Procesi (CH–DP) equation. Some new bright and dark soliton solutions are obtained. Also, some new doubly periodic solutions in the form of Jacobi elliptic functions and Weierstrass elliptic functions are obtained.

scholarly journals Generalized Jacobi Elliptic Function Solution to a Class of Nonlinear Schrödinger-Type Equations

2011 ◽  
Vol 2011 ◽  
pp. 1-11 ◽  
Author(s):  
Zeid I. A. Al-Muhiameed ◽  
Emad A.-B. Abdel-Salam
Keyword(s):  
Elliptic Function ◽  
Traveling Wave ◽  
Function Method ◽  
Nonlinear Partial Differential Equations ◽  
Traveling Wave Solutions ◽  
Jacobi Elliptic Function ◽  
Nonlinear Schrödinger ◽  
Function Solution ◽  
Balance Principle ◽  
Wave Solutions

With the help of the generalized Jacobi elliptic function, an improved Jacobi elliptic function method is used to construct exact traveling wave solutions of the nonlinear partial differential equations in a unified way. A class of nonlinear Schrödinger-type equations including the generalized Zakharov system, the Rangwala-Rao equation, and the Chen-Lee-Lin equation are investigated, and the exact solutions are derived with the aid of the homogenous balance principle.

scholarly journals Exact Solutions of the Kudryashov-Sinelshchikov Equation Using the MultipleG′/G-Expansion Method

2013 ◽  
Vol 2013 ◽  
pp. 1-7 ◽  
Author(s):  
Yinghui He ◽  
Shaolin Li ◽  
Yao Long
Keyword(s):  
Exact Solutions ◽  
Traveling Wave ◽  
Traveling Wave Solutions ◽  
Elliptic Functions ◽  
Rational Functions ◽  
Expansion Method ◽  
Jacobi Elliptic Functions ◽  
Hyperbolic Functions ◽  
Exact Traveling Wave Solutions ◽  
Wave Solutions

Exact traveling wave solutions of the Kudryashov-Sinelshchikov equation are studied by theG′/G-expansion method and its variants. The solutions obtained include the form of Jacobi elliptic functions, hyperbolic functions, and trigonometric and rational functions. Many new exact traveling wave solutions can easily be derived from the general results under certain conditions. These methods are effective, simple, and many types of solutions can be obtained at the same time.

THE EXACT TRAVELING WAVE SOLUTIONS AND THEIR BIFURCATIONS IN THE GARDNER AND GARDNER–KP EQUATIONS

2012 ◽  
Vol 22 (05) ◽  
pp. 1250126 ◽  
Author(s):  
FANG YAN ◽  
CUNCAI HUA ◽  
HAIHONG LIU ◽  
ZENGRONG LIU
Keyword(s):  
Traveling Wave ◽  
Function Method ◽  
Traveling Wave Solutions ◽  
Analytic Solutions ◽  
Auxiliary Function ◽  
Kp Equation ◽  
Gardner Equation ◽  
Exact Traveling Wave Solutions ◽  
Wave Solutions ◽  
Kp Equations

By using the method of dynamical systems, this paper studies the exact traveling wave solutions and their bifurcations in the Gardner equation. Exact parametric representations of all wave solutions as well as the explicit analytic solutions are given. Moreover, several series of exact traveling wave solutions of the Gardner–KP equation are obtained via an auxiliary function method.

Abundant soliton solutions for the Hirota–Maccari equation via the generalized exponential rational function method

2019 ◽  
Vol 33 (09) ◽  
pp. 1950106 ◽  
Author(s):  
Behzad Ghanbari
Keyword(s):  
Exact Solutions ◽  
Rational Function ◽  
Traveling Wave ◽  
Function Method ◽  
Soliton Solutions ◽  
Traveling Wave Solutions ◽  
Graphical Interpretation ◽  
Wave Solutions ◽  
Complex Models ◽  
Exponential Rational Function Method

In this paper, some new traveling wave solutions to the Hirota–Maccari equation are constructed with the help of the newly introduced method called generalized exponential rational function method. Several families of exact solutions are found corresponding to the equation. To the best of our knowledge, these solutions are new, and have never been addressed in the literature. The graphical interpretation of the solutions is also depicted. Moreover, it is contemplated that the proposed technique can also be employed to another sort of complex models.

EXACT TRAVELING WAVE SOLUTIONS OF A HIGHER-DIMENSIONAL NONLINEAR EVOLUTION EQUATION

2010 ◽  
Vol 24 (10) ◽  
pp. 1011-1021 ◽  
Author(s):  
JONU LEE ◽  
RATHINASAMY SAKTHIVEL ◽  
LUWAI WAZZAN
Keyword(s):  
Traveling Wave ◽  
Function Method ◽  
Soliton Solutions ◽  
Traveling Wave Solutions ◽  
Periodic Wave ◽  
Jacobi Elliptic Function ◽  
Periodic Wave Solutions ◽  
Exact Traveling Wave Solutions ◽  
Wave Solutions ◽  
Higher Dimensional

The exact traveling wave solutions of (4 + 1)-dimensional nonlinear Fokas equation is obtained by using three distinct methods with symbolic computation. The modified tanh–coth method is implemented to obtain single soliton solutions whereas the extended Jacobi elliptic function method is applied to derive doubly periodic wave solutions for this higher-dimensional integrable equation. The Exp-function method gives generalized wave solutions with some free parameters. It is shown that soliton solutions and triangular solutions can be established as the limits of the Jacobi doubly periodic wave solutions.

scholarly journals New Traveling Wave Solutions of the Higher Dimensional Nonlinear Partial Differential Equation by the Exp-Function Method

2012 ◽  
Vol 2012 ◽  
pp. 1-14 ◽  
Author(s):  
Hasibun Naher ◽  
Farah Aini Abdullah ◽  
M. Ali Akbar
Keyword(s):  
Differential Equation ◽  
Partial Differential Equation ◽  
Analytical Solutions ◽  
Traveling Wave ◽  
Function Method ◽  
Nonlinear Partial Differential Equation ◽  
Traveling Wave Solutions ◽  
Partial Differential ◽  
Wave Solutions ◽  
Higher Dimensional

We construct new analytical solutions of the (3+1)-dimensional modified KdV-Zakharov-Kuznetsev equation by the Exp-function method. Plentiful exact traveling wave solutions with arbitrary parameters are effectively obtained by the method. The obtained results show that the Exp-function method is effective and straightforward mathematical tool for searching analytical solutions with arbitrary parameters of higher-dimensional nonlinear partial differential equation.

An analytical method for soliton solutions of perturbed Schrödinger’s equation with quadratic-cubic nonlinearity

2019 ◽  
Vol 33 (03) ◽  
pp. 1950018 ◽  
Author(s):  
Behzad Ghanbari ◽  
Nauman Raza
Keyword(s):  
Traveling Wave ◽  
Function Method ◽  
Soliton Solutions ◽  
Traveling Wave Solutions ◽  
Cubic Nonlinearity ◽  
Nonlinear Media ◽  
Schrödinger’S Equation ◽  
Wave Solutions ◽  
First Time ◽  
Schrödinger's Equation

In this study, we acquire some new exact traveling wave solutions to the nonlinear Schrödinger’s equation in the presence of Hamiltonian perturbations. The compendious integration tool, generalized exponential rational function method (GERFM), is utilized in the presence of quadratic-cubic nonlinear media. The obtained results depict the efficiency of the proposed scheme and are being reported for the first time.

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