A NEW GENERAL ALGEBRAIC METHOD WITH SYMBOLIC COMPUTATION TO CONSTRUCT NEW TRAVELING SOLUTION FOR THE (1 +1)-DIMENSIONAL DISPERSIVE LONG WAVE EQUATION

2005 ◽  
Vol 16 (07) ◽  
pp. 1107-1119 ◽  
Author(s):  
YONG CHEN
Keyword(s):  
Wave Equation ◽  
Symbolic Computation ◽  
Evolution Equations ◽  
Traveling Wave Solutions ◽  
Algebraic Method ◽  
Periodic Wave ◽  
Nonlinear Evolution Equations ◽  
Rational Form ◽  
Wave Solutions ◽  
Long Wave

A new algebraic method, named Riccati equation rational expansion (RERE) method, is devised for constructing multiple traveling wave solutions for nonlinear evolution equations (NEEs). With the aid of symbolic computation, we choose (1 +1)-dimensional dispersive long wave equation (DLWE) to illustrate our method. As a result, we obtain many types of solutions including rational form solitary wave solutions, triangular periodic wave solutions and rational wave solutions.

CONSTRUCTING FAMILIES TRAVELING WAVE SOLUTIONS IN TERMS OF SPECIAL FUNCTION FOR THE ASYMMETRIC NIZHNIK–NOVIKOV–VESSELOV EQUATION

2004 ◽  
Vol 15 (04) ◽  
pp. 595-606 ◽  
Author(s):  
YONG CHEN ◽  
QI WANG
Keyword(s):  
Traveling Wave ◽  
Evolution Equations ◽  
Special Function ◽  
Nonlinear Evolution ◽  
Traveling Wave Solutions ◽  
Algebraic Method ◽  
Nonlinear Evolution Equations ◽  
Symbolic System ◽  
Wave Solutions

By means of a more general ansatz and the computerized symbolic system Maple, a generalized algebraic method to uniformly construct solutions in terms of special function of nonlinear evolution equations (NLEEs) is presented. We not only successfully recover the previously-known traveling wave solutions found by Fan's method, but also obtain some general traveling wave solutions in terms of the special function for the asymmetric Nizhnik–Novikov–Vesselov equation.

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)  

scholarly journals Traveling Wave Solutions of the Kadomtsev-Petviashvili-Benjamin-Bona-Mahony Equation

2014 ◽  
Vol 2014 ◽  
pp. 1-9
Author(s):  
Zhengyong Ouyang
Keyword(s):  
Traveling Wave ◽  
Evolution Equations ◽  
Nonlinear Evolution ◽  
Traveling Wave Solutions ◽  
Periodic Wave ◽  
Nonlinear Evolution Equations ◽  
Periodic Waves ◽  
Bifurcation Method ◽  
Periodic Wave Solutions ◽  
Wave Solutions

We use bifurcation method of dynamical systems to study exact traveling wave solutions of a nonlinear evolution equation. We obtain exact explicit expressions of bell-shaped solitary wave solutions involving more free parameters, and some existing results are corrected and improved. Also, we get some new exact periodic wave solutions in parameter forms of the Jacobian elliptic function. Further, we find that the bell-shaped waves are limits of the periodic waves in some sense. The results imply that we can deduce bell-shaped waves from periodic waves for some nonlinear evolution equations.

scholarly journals Exact Solutions to the Sharma-Tasso-Olver Equation by Using ImprovedG′/G-Expansion Method

2013 ◽  
Vol 2013 ◽  
pp. 1-6 ◽  
Author(s):  
Yinghui He ◽  
Shaolin Li ◽  
Yao Long
Keyword(s):  
Evolution Equations ◽  
Traveling Wave Solutions ◽  
Expansion Method ◽  
Periodic Wave ◽  
Nonlinear Evolution Equations ◽  
Hyperbolic Function ◽  
Periodic Traveling Waves ◽  
Wave Solutions ◽  
Hyperbolic Function Solutions ◽  
Trigonometric Function Solutions

This paper is concerned with a double nonlinear dispersive equation: the Sharma-Tasso-Olver equation. We propose an improvedG′/G-expansion method which is employed to investigate the solitary and periodic traveling waves of this equation. As a result, some new traveling wave solutions involving hyperbolic functions, the trigonometric functions, are obtained. When the parameters are taken as special values, the solitary wave solutions are derived from the hyperbolic function solutions, and the periodic wave solutions are derived from the trigonometric function solutions. The improvedG′/G-expansion method is straightforward, concise and effective and can be applied to other nonlinear evolution equations in mathematical physics.

scholarly journals Bifurcation of Traveling Wave Solutions for (2+1)-Dimensional Nonlinear Models Generated by the Jaulent-Miodek Hierarchy

2015 ◽  
Vol 2015 ◽  
pp. 1-14
Author(s):  
Yanping Ran ◽  
Jing Li ◽  
Xin Li ◽  
Zheng Tian
Keyword(s):  
Evolution Equations ◽  
Nonlinear Models ◽  
Traveling Wave Solutions ◽  
Periodic Wave ◽  
Nonlinear Evolution Equations ◽  
Phase Portraits ◽  
Wave Solutions ◽  
System Method ◽  
Dynamical System Method ◽  
Planar Dynamical Systems

Four (2+1)-dimensional nonlinear evolution equations, generated by the Jaulent-Miodek hierarchy, are investigated by the bifurcation method of planar dynamical systems. The bifurcation regions in different subsets of the parameters space are obtained. According to the different phase portraits in different regions, we obtain kink (antikink) wave solutions, solitary wave solutions, and periodic wave solutions for the third of these models by dynamical system method. Furthermore, the explicit exact expressions of these bounded traveling waves are obtained. All these wave solutions obtained are characterized by distinct physical structures.

Explicit, periodic and dispersive soliton solutions to the conformable time-fractional Wu–Zhang system

2021 ◽  
pp. 2150417
Author(s):  
Kalim U. Tariq ◽  
Mostafa M. A. Khater ◽  
Muhammad Younis
Keyword(s):  
Fractional Derivative ◽  
Traveling Wave ◽  
Evolution Equations ◽  
Nonlinear Evolution ◽  
Soliton Solutions ◽  
Traveling Wave Solutions ◽  
Nonlinear Evolution Equations ◽  
Conformable Fractional Derivative ◽  
Physical Behavior ◽  
Wave Solutions

In this paper, some new traveling wave solutions to the conformable time-fractional Wu–Zhang system are constructed with the help of the extended Fan sub-equation method. The conformable fractional derivative is employed to transform the fractional form of the system into ordinary differential system with an integer order. Some distinct types of figures are sketched to illustrate the physical behavior of the obtained solutions. The power and effective of the used method is shown and its ability for applying different forms of nonlinear evolution equations is also verified.

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