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 Generalized Hyperbolic Function Solution to a Class of Nonlinear Schrödinger-Type Equations

2012 ◽  
Vol 2012 ◽  
pp. 1-15 ◽  
Author(s):  
Zeid I. A. Al-Muhiameed ◽  
Emad A.-B. Abdel-Salam
Keyword(s):  
Traveling Wave ◽  
Nonlinear Partial Differential Equations ◽  
Traveling Wave Solutions ◽  
Hyperbolic Function ◽  
Hyperbolic Functions ◽  
Nonlinear Schrödinger ◽  
Function Solution ◽  
Balance Principle ◽  
Hyperbolic Function Solution ◽  
Differential Equation Method

With the help of the generalized hyperbolic function, the subsidiary ordinary differential equation method is improved and proposed 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-Liu equation are investigated and the exact solutions are derived with the aid of the homogenous balance principle and generalized hyperbolic functions. We study the effect of the generalized hyperbolic function parameterspandqin the obtained solutions by using the computer simulation.

EXACT TRAVELING WAVE SOLUTIONS OF THE GENERALIZED HIROTA–SATSUMA COUPLED KdV EQUATION

2010 ◽  
Vol 24 (22) ◽  
pp. 4333-4355 ◽  
Author(s):  
ZHU LI
Keyword(s):  
Elliptic Function ◽  
Traveling Wave ◽  
Function Method ◽  
Kdv Equation ◽  
Traveling Wave Solutions ◽  
Jacobi Elliptic Function ◽  
Exact Traveling Wave Solutions ◽  
Wave Solutions ◽  
Coupled Kdv Equation ◽  
Jacobi Elliptic Function Method

Exact traveling wave solutions of the generalized Hirota–Satsuma coupled KdV equation are obtained by the generalized Jacobi elliptic function method.

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.

Multiple wave solutions for nonlinear burgers equations using the multiple exp-function method

2021 ◽  
pp. 2150149
Author(s):  
Khaled A. Gepreel ◽  
E. M. E. Zayed
Keyword(s):  
Traveling Wave ◽  
Software Package ◽  
Function Method ◽  
Nonlinear Partial Differential Equations ◽  
Soliton Solutions ◽  
Traveling Wave Solutions ◽  
Mathematical Physics ◽  
Multiple Wave ◽  
Wave Solutions ◽  
The One

In this paper, we use the multiple exp-function method to explicity present traveling wave solutions, double-traveling wave (DTW) solutions and triple-traveling wave solutions (TWs) which include one-soliton, double-soliton and triple-soliton solutions for nonlinear partial differential equations (NPDEs) via, the (2+1)-dimensional and (3+1)-dimensional nonlinear Burgers PDEs in mathematical physics. In this work, we build some series of straightforward and new solutions successfully with the help of a computerized symbol computational software package like Maple or Mathematica. We will make some drawings in some cases with specific values for the relevant parameters for each obtained solutions such as the one-traveling wave solutions, double-traveling wave solutions and TWs. This method is efficient and powerful in solving a wide class of NPDEs.

scholarly journals Solitons for the modified $(2 + 1)$-dimensional Konopelchenko–Dubrovsky equations

2019 ◽  
Vol 2019 (1) ◽  
Author(s):  
Xiumei Lyu ◽  
Wei Gu
Keyword(s):  
Elliptic Function ◽  
Traveling Wave ◽  
Traveling Wave Solutions ◽  
High Order ◽  
Auxiliary Equation ◽  
Jacobi Elliptic Function ◽  
Equation Approach ◽  
Exact Traveling Wave Solutions ◽  
Wave Solutions ◽  
Triangular Function

Abstract In the paper, we consider the modified $(2 + 1)$ ( 2 + 1 ) -dimensional Konopelchenko–Dubrovsky equations which possess high order nonlinear terms. Under the aid of Maple, we derive the exact traveling wave solutions of the mKDs by the auxiliary equation approach. Under some special conditions, Jacobi elliptic function solutions, degenerated triangular function solutions, and solitons for the mKD equations are constructed.

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 An Extended Auxiliary Function Method and Its Application in mKdV Equation

2013 ◽  
Vol 2013 ◽  
pp. 1-7
Author(s):  
Yafeng Xiao ◽  
Haili Xue ◽  
Hongqing Zhang
Keyword(s):  
Partial Differential Equations ◽  
Differential Equations ◽  
Traveling Wave ◽  
Function Method ◽  
Nonlinear Partial Differential Equations ◽  
Traveling Wave Solutions ◽  
Mkdv Equation ◽  
Auxiliary Function ◽  
Exact Traveling Wave Solutions ◽  
Wave Solutions

An extended auxiliary function method is presented for constructing exact traveling wave solutions to nonlinear partial differential equations. The main idea of this method is to take full advantage of the solutions to the elliptic equation to construct exact traveling wave solutions for nonlinear partial differential equations. mKdV equation is chosen to illustrate the application of the extended auxiliary function method. Consequently, more new exact traveling wave solutions are derived that are not obtained by the previously known methods.

scholarly journals Symbolic Computation and the Extended Hyperbolic Function Method for Constructing Exact Traveling Solutions of Nonlinear PDEs

2012 ◽  
Vol 2012 ◽  
pp. 1-19 ◽  
Author(s):  
Huang Yong ◽  
Shang Yadong ◽  
Yuan Wenjun
Keyword(s):  
Traveling Wave ◽  
Function Method ◽  
Nonlinear Partial Differential Equations ◽  
Traveling Wave Solutions ◽  
Boussinesq Equations ◽  
Nonlinear Pdes ◽  
Hyperbolic Function ◽  
Wave Solutions ◽  
Hyperbolic Function Method ◽  
Periodic Traveling Wave

On the basis of the computer symbolic system Maple and the extended hyperbolic function method, we develop a more mathematically rigorous and systematic procedure for constructing exact solitary wave solutions and exact periodic traveling wave solutions in triangle form of various nonlinear partial differential equations that are with physical backgrounds. Compared with the existing methods, the proposed method gives new and more general solutions. More importantly, the method provides a straightforward and effective algorithm to obtain abundant explicit and exact particular solutions for large nonlinear mathematical physics equations. We apply the presented method to two variant Boussinesq equations and give a series of exact explicit traveling wave solutions that have some more general forms. So consequently, the efficiency and the generality of the proposed method are demonstrated.

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