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.

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 The Classification of Single Traveling Wave Solutions for the Fractional Coupled Nonlinear SchrÖdinger equation

2021 ◽  
Author(s):  
Lu Tang ◽  
Shanpeng Chen
Keyword(s):  
Schrödinger Equation ◽  
Nonlinear Schrödinger Equation ◽  
Traveling Wave ◽  
Schrodinger Equation ◽  
Traveling Wave Solutions ◽  
Nonlinear Schrodinger Equation ◽  
Hyperbolic Function ◽  
Nonlinear Schrödinger ◽  
Coupled Nonlinear Schrödinger Equation ◽  
Wave Solutions

Abstract The main purpose of this paper is to study the single traveling wave solutions of the fractional coupled nonlinear SchrÖdinger equation. By using the complete discriminant system method and computer algebra with symbolic computation, a series of new single traveling wave solutions are obtained, which include trigonometric function solutions, Jacobi elliptic function solutions, hyperbolic function solutions, solitary wave solutions and rational function solutions. In order to further explain the propagation of the fractional coupled nonlinear Schr\"{o}dinger equation in nonlinear optics, two-dimensional and three-dimensional graphs are drawn.

scholarly journals Traveling Wave Solutions of Two Nonlinear Wave Equations by (G′/G)-Expansion Method

2018 ◽  
Vol 2018 ◽  
pp. 1-8 ◽  
Author(s):  
Yazhou Shi ◽  
Xiangpeng Li ◽  
Ben-gong Zhang
Keyword(s):  
Nonlinear Wave ◽  
Traveling Wave ◽  
Wave Equations ◽  
Traveling Wave Solutions ◽  
Expansion Method ◽  
Nonlinear Wave Equations ◽  
Hyperbolic Function ◽  
Exact Traveling Wave Solutions ◽  
Function Solution ◽  
Wave Solutions

We employ the (G′/G)-expansion method to seek exact traveling wave solutions of two nonlinear wave equations—Padé-II equation and Drinfel’d-Sokolov-Wilson (DSW) equation. As a result, hyperbolic function solution, trigonometric function solution, and rational solution with general parameters are obtained. The interesting thing is that the exact solitary wave solutions and new exact traveling wave solutions can be obtained when the special values of the parameters are taken. Comparing with other methods, the method used in this paper is very direct. The (G′/G)-expansion method presents wide applicability for handling nonlinear wave equations.

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.

scholarly journals Study on the applications of two analytical methods for the construction of traveling wave solutions of the modified equal width equation

Open Physics ◽  
2020 ◽  
Vol 18 (1) ◽  
pp. 1003-1010
Author(s):  
Asıf Yokuş ◽  
Hülya Durur ◽  
Taher A. Nofal ◽  
Hanaa Abu-Zinadah ◽  
Münevver Tuz ◽  
...  
Keyword(s):  
Traveling Wave ◽  
Analytical Methods ◽  
Nonlinear Evolution ◽  
Nonlinear Partial Differential Equations ◽  
Traveling Wave Solutions ◽  
Dark Soliton ◽  
Mathematical Tool ◽  
Symbolic Calculation ◽  
Advantages And Disadvantages ◽  
Wave Solutions

Abstract In this article, the Sinh–Gordon function method and sub-equation method are used to construct traveling wave solutions of modified equal width equation. Thanks to the proposed methods, trigonometric soliton, dark soliton, and complex hyperbolic solutions of the considered equation are obtained. Common aspects, differences, advantages, and disadvantages of both analytical methods are discussed. It has been shown that the traveling wave solutions produced by both analytical methods with different base equations have different properties. 2D, 3D, and contour graphics are offered for solutions obtained by choosing appropriate values of the parameters. To evaluate the feasibility and efficacy of these techniques, a nonlinear evolution equation was investigated, and with the help of symbolic calculation, these methods have been shown to be a powerful, reliable, and effective mathematical tool for the solution of nonlinear partial differential equations.

scholarly journals Fundamental solutions for the long–short-wave interaction system

Open Physics ◽  
2020 ◽  
Vol 18 (1) ◽  
pp. 1093-1099
Author(s):  
Mustafa Inc ◽  
Samia Zaki Hassan ◽  
Mahmoud Abdelrahman ◽  
Reem Abdalaziz Alomair ◽  
Yu-Ming Chu
Keyword(s):  
Fluid Mechanics ◽  
Traveling Wave ◽  
Inverse Method ◽  
Nonlinear Partial Differential Equations ◽  
Traveling Wave Solutions ◽  
Wave Interaction ◽  
Fundamental Solutions ◽  
Short Wave ◽  
Wave Solutions ◽  
Physical Environments

Abstract In this article, the system for the long–short-wave interaction (LS) system is considered. In order to construct some new traveling wave solutions, He’s semi-inverse method is implemented. These solutions may be applicable for some physical environments, such as physics and fluid mechanics. These new solutions show that the proposed method is easy to apply and the proposed technique is a very powerful tool to solve many other nonlinear partial differential equations in applied science.

scholarly journals Approximate Traveling Wave Solutions of Coupled Whitham-Broer-Kaup Shallow Water Equations by Homotopy Analysis Method

2008 ◽  
Vol 2008 ◽  
pp. 1-8 ◽  
Author(s):  
M. M. Rashidi ◽  
D. D. Ganji ◽  
S. Dinarvand
Keyword(s):  
Exact Solutions ◽  
Shallow Water ◽  
Homotopy Analysis Method ◽  
Shallow Water Equations ◽  
Traveling Wave ◽  
Nonlinear Partial Differential Equations ◽  
Traveling Wave Solutions ◽  
Analysis Method ◽  
Wave Solutions ◽  
Homotopy Analysis

The homotopy analysis method (HAM) is applied to obtain the approximate traveling wave solutions of the coupled Whitham-Broer-Kaup (WBK) equations in shallow water. Comparisons are made between the results of the proposed method and exact solutions. The results show that the homotopy analysis method is an attractive method in solving the systems of nonlinear partial differential equations.

scholarly journals The Improved Generalized tanh-coth Method Applied to Sixth-Order Solitary Wave Equation

2017 ◽  
Vol 2017 ◽  
pp. 1-11
Author(s):  
M. Torvattanabun ◽  
J. Simmapim ◽  
D. Saennuad ◽  
T. Somaumchan
Keyword(s):  
Wave Equation ◽  
Solitary Wave ◽  
Nonlinear Partial Differential Equations ◽  
Traveling Wave Solutions ◽  
Rational Functions ◽  
Sixth Order ◽  
Mathematical Tool ◽  
Trigonometric Functions ◽  
Hyperbolic Functions ◽  
New Exact Solutions

The improved generalized tanh-coth method is used in nonlinear sixth-order solitary wave equation. This method is a powerful and advantageous mathematical tool for establishing abundant new traveling wave solutions of nonlinear partial differential equations. The new exact solutions consisted of trigonometric functions solutions, hyperbolic functions solutions, exponential functions solutions, and rational functions solutions. The numerical results were obtained with the aid of Maple.

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