scholarly journals Solitary Wave Solutions of Some Coupled Nonlinear Evolution Equations

2014 ◽  
Vol 6 (2) ◽  
pp. 273-284 ◽  
Author(s):  
K. Khan ◽  
M. A. Akbar
Keyword(s):  
Solitary Wave ◽  
Traveling Wave ◽  
Evolution Equations ◽  
Nonlinear Evolution ◽  
Traveling Wave Solutions ◽  
Mathematical Physics ◽  
Nonlinear Evolution Equations ◽  
Solitary Wave Solutions ◽  
Wave Solutions ◽  
Long Wave

In this article, the modified simple equation (MSE) method has been executed to find the traveling wave solutions of the coupled (1+1)-dimensional Broer-Kaup (BK) equations and the dispersive long wave (DLW) equations. The efficiency of the method for finding exact solutions has been demonstrated. It has been shown that the method is direct, effective and can be used for many other nonlinear evolution equations (NLEEs) in mathematical physics. Moreover, this procedure reduces the large volume of calculations.  Keywords: MSE method; NLEE; BK equations; DLW equations; Solitary wave solutions. © 2014 JSR Publications. ISSN: 2070-0237 (Print); 2070-0245 (Online). All rights reserved. doi: http://dx.doi.org/10.3329/jsr.v6i2.16671 J. Sci. Res. 6 (2), 273-284 (2014)  

EXACT TRAVELING WAVE SOLUTIONS FOR SOME NONLINEAR EVOLUTION EQUATIONS WITH NONLINEAR TERMS OF ANY ORDER

2003 ◽  
Vol 14 (01) ◽  
pp. 99-112 ◽  
Author(s):  
YONG CHEN ◽  
BIAO LI ◽  
HONG-QING ZHANG
Keyword(s):  
Solitary Wave ◽  
Traveling Wave ◽  
Evolution Equations ◽  
Nonlinear Evolution ◽  
Traveling Wave Solutions ◽  
Nonlinear Evolution Equations ◽  
Solitary Wave Solutions ◽  
Rational Solutions ◽  
Exact Traveling Wave Solutions ◽  
Wave Solutions

In this paper, we improved the tanh method by means of a proper transformation and general ansätz. Using the improved method, with the aid of Mathematica™, we consider some nonlinear evolution equations with nonlinear terms of any order. As a result, rich explicit exact traveling wave solutions for these equations, which contain kink profile solitary wave solutions, bell profile solitary wave solutions, rational solutions, periodic solutions, and combined formal solutions, are obtained.

scholarly journals The Modified Simple Equation Method for Exact and Solitary Wave Solutions of Nonlinear Evolution Equation: The GZK-BBM Equation and Right-Handed Noncommutative Burgers Equations

2013 ◽  
Vol 2013 ◽  
pp. 1-5 ◽  
Author(s):  
Kamruzzaman Khan ◽  
M. Ali Akbar ◽  
Norhashidah Hj. Mohd. Ali
Keyword(s):  
Solitary Wave ◽  
Traveling Wave ◽  
Nonlinear Evolution ◽  
Traveling Wave Solutions ◽  
Mathematical Physics ◽  
Simple Equation ◽  
Solitary Wave Solutions ◽  
Wave Solutions ◽  
Modified Simple Equation Method ◽  
Bbm Equation

The modified simple equation method is significant for finding the exact traveling wave solutions of nonlinear evolution equations (NLEEs) in mathematical physics. In this paper, we bring in the modified simple equation (MSE) method for solving NLEEs via the Generalized Zakharov-Kuznetsov-Benjamin-Bona-Mahony (GZK-BBM) equation and the right-handed noncommutative Burgers' (nc-Burgers) equations and achieve the exact solutions involving parameters. When the parameters are taken as special values, the solitary wave solutions are originated from the traveling wave solutions. It is established that the MSE method offers a further influential mathematical tool for constructing the exact solutions of NLEEs in mathematical physics.

scholarly journals Further Results about Traveling Wave Exact Solutions of the Drinfeld-Sokolov Equations

2013 ◽  
Vol 2013 ◽  
pp. 1-6 ◽  
Author(s):  
Fu Zhang ◽  
Jian-ming Qi ◽  
Wen-jun Yuan
Keyword(s):  
Exact Solutions ◽  
Traveling Wave ◽  
Evolution Equations ◽  
Nonlinear Evolution ◽  
Traveling Wave Solutions ◽  
Mathematical Physics ◽  
Nonlinear Evolution Equations ◽  
Complex Method ◽  
Wave Solutions ◽  
Periodic Traveling Wave

We employ the complex method to obtain all meromorphic exact solutions of complex Drinfeld-Sokolov equations (DS system of equations). The idea introduced in this paper can be applied to other nonlinear evolution equations. Our results show that all constant and simply periodic traveling wave exact solutions of the equations (DS) are solitary wave solutions, the complex method is simpler than other methods and there exist simply periodic solutionsvs,3(z)which are not only new but also not degenerated successively by the elliptic function solutions. We believe that this method should play an important role for finding exact solutions in the mathematical physics. For these new traveling wave solutions, we give some computer simulations to illustrate our main results.

Solitary wave solutions to nonlinear evolution equations in mathematical physics

Pramana ◽  
2014 ◽  
Vol 83 (4) ◽  
pp. 457-471 ◽  
Author(s):  
ANWAR JA’AFAR MOHAMAD JAWAD ◽  
M MIRZAZADEH ◽  
ANJAN BISWAS
Keyword(s):  
Solitary Wave ◽  
Evolution Equations ◽  
Nonlinear Evolution ◽  
Mathematical Physics ◽  
Nonlinear Evolution Equations ◽  
Solitary Wave Solutions ◽  
Wave Solutions

scholarly journals Abundant Explicit Solutions to Fractional Order Nonlinear Evolution Equations

2021 ◽  
Vol 2021 ◽  
pp. 1-16
Author(s):  
M. Ayesha Khatun ◽  
Mohammad Asif Arefin ◽  
M. Hafiz Uddin ◽  
Mustafa Inc
Keyword(s):  
Solitary Wave ◽  
Traveling Wave ◽  
Evolution Equations ◽  
Traveling Wave Solutions ◽  
Nonlinear Evolution Equations ◽  
Mathematical Tool ◽  
Solitary Wave Solutions ◽  
Dark Solitons ◽  
Wave Solutions ◽  
Liouville Derivative

We utilize the modified Riemann–Liouville derivative sense to develop careful arrangements of time-fractional simplified modified Camassa–Holm (MCH) equations and generalized (3 + 1)-dimensional time-fractional Camassa–Holm–Kadomtsev–Petviashvili (gCH-KP) through the potential double G ′ / G , 1 / G -expansion method (DEM). The mentioned equations describe the role of dispersion in the formation of patterns in liquid drops ensued in plasma physics, optical fibers, fluid flow, fission and fusion phenomena, acoustics, control theory, viscoelasticity, and so on. A generalized fractional complex transformation is appropriately used to change this equation to an ordinary differential equation; thus, many precise logical arrangements are acquired with all the freer parameters. At the point when these free parameters are taken as specific values, the traveling wave solutions are transformed into solitary wave solutions expressed by the hyperbolic, the trigonometric, and the rational functions. The physical significance of the obtained solutions for the definite values of the associated parameters is analyzed graphically with 2D, 3D, and contour format. Scores of solitary wave solutions are obtained such as kink type, periodic wave, singular kink, dark solitons, bright-dark solitons, and some other solitary wave solutions. It is clear to scrutinize that the suggested scheme is a reliable, competent, and straightforward mathematical tool to discover closed form traveling wave solutions.

scholarly journals The Generalized Projective Riccati Equations Method for Solving Nonlinear Evolution Equations in Mathematical Physics

2014 ◽  
Vol 2014 ◽  
pp. 1-10 ◽  
Author(s):  
E. M. E. Zayed ◽  
K. A. E. Alurrfi
Keyword(s):  
Traveling Wave ◽  
Evolution Equations ◽  
Burgers Equation ◽  
Nonlinear Evolution ◽  
Traveling Wave Solutions ◽  
Riccati Equations ◽  
Mathematical Physics ◽  
Nonlinear Evolution Equations ◽  
Exact Traveling Wave Solutions ◽  
Wave Solutions

We apply the generalized projective Riccati equations method to find the exact traveling wave solutions of some nonlinear evolution equations with any-order nonlinear terms, namely, the nonlinear Pochhammer-Chree equation, the nonlinear Burgers equation and the generalized, nonlinear Zakharov-Kuznetsov equation. This method presents wider applicability for handling many other nonlinear evolution equations in mathematical physics.

scholarly journals Exact Solutions to the (2+1)-Dimensional Boussinesq Equation via exp(?(?))-Expansion Method

2015 ◽  
Vol 7 (3) ◽  
pp. 1-10 ◽  
Author(s):  
M. N. Alam ◽  
M. G. Hafez ◽  
M. A. Akbar ◽  
H. -O. -Roshid
Keyword(s):  
Traveling Wave ◽  
Evolution Equations ◽  
Boussinesq Equation ◽  
Nonlinear Evolution ◽  
Traveling Wave Solutions ◽  
Rational Functions ◽  
Expansion Method ◽  
Mathematical Physics ◽  
Nonlinear Evolution Equations ◽  
Wave Solutions

The exp(?(?))-expansion method is applied to find exact traveling wave solutions to the (2+1)-dimensional Boussinesq equation which is an important equation in mathematical physics. The traveling wave solutions are expressed in terms of the exponential functions, the hyperbolic functions, the trigonometric functions and the rational functions. The procedure is simple, direct and constructive without the help of a computer algebra system. The applied method will be used in further works to establish more new solutions for other kinds of nonlinear evolution equations arising in mathematical physics and engineering.

scholarly journals Mathematical modeling of DNA vibrational dynamics and its solitary wave solutions

2018 ◽  
Vol 64 (6) ◽  
pp. 590 ◽  
Author(s):  
Reza Abazari ◽  
Shabnam Jamshidzadeh ◽  
Gangwei Wang
Keyword(s):  
Mathematical Modeling ◽  
Traveling Wave ◽  
Evolution Equations ◽  
Nonlinear Evolution ◽  
Traveling Wave Solutions ◽  
Nonlinear Evolution Equations ◽  
Solitary Wave Solutions ◽  
Vibrational Dynamics ◽  
Wave Solutions ◽  
Vibration Dynamics

In this work, the traveling wave solutions of a mathematical modeling of DNA vibration dynamics proposed by Peyrard-Bishop, that takes into consideration the inclusion of nonlinear interaction between adjacent displacements along the Hydrogen bonds, is investigated by both $(G'/G)$-expansion and $F$-expansion methods. Using these methods, some new explicit forms of traveling wave solutions of present nonlinear equation are given. The methods come in to be easier and faster by means of a symbolic computation and yield powerful mathematical tools for solving nonlinear evolution equations in many branches of sciences, especially physics, biology and etc.

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