scholarly journals Travelling Wave Solutions of the Schrödinger-Boussinesq System

2012 ◽  
Vol 2012 ◽  
pp. 1-11 ◽  
Author(s):  
Adem Kılıcman ◽  
Reza Abazari
Keyword(s):  
Soliton Solutions ◽  
Expansion Method ◽  
Travelling Wave ◽  
Trigonometric Function ◽  
Boussinesq System ◽  
Hyperbolic Function ◽  
Travelling Wave Solutions ◽  
Physical Problems ◽  
Hyperbolic Function Solutions ◽  
Trigonometric Function Solutions

We establish exact solutions for the Schrödinger-Boussinesq Systemiut+uxx−auv=0,vtt−vxx+vxxxx−b(|u|2)xx=0, whereaandbare real constants. The (G′/G)-expansion method is used to construct exact periodic and soliton solutions of this equation. Our work is motivated by the fact that the (G′/G)-expansion method provides not only more general forms of solutions but also periodic and solitary waves. As a result, hyperbolic function solutions and trigonometric function solutions with parameters are obtained. These solutions may be important and of significance for the explanation of some practical physical problems.

Non-TravellingWave Solutions of the (2+1)-Dimensional Dispersive Long Wave System

2009 ◽  
Vol 64 (9-10) ◽  
pp. 540-552
Author(s):  
Mamdouh M. Hassan
Keyword(s):  
Expansion Method ◽  
Travelling Wave ◽  
Hyperbolic Function ◽  
Wave System ◽  
Jacobi Elliptic Function ◽  
Travelling Wave Solutions ◽  
Rational Solutions ◽  
Long Wave ◽  
Hyperbolic Function Solutions ◽  
Trigonometric Function Solutions

With the aid of symbolic computation and the extended F-expansion method, we construct more general types of exact non-travelling wave solutions of the (2+1)-dimensional dispersive long wave system. These solutions include single and combined Jacobi elliptic function solutions, rational solutions, hyperbolic function solutions, and trigonometric function solutions.

scholarly journals New travelling wave solutions for fractional regularized long-wave equation and fractional coupled Nizhnik-Novikov-Veselov equation

2017 ◽  
Vol 8 (1) ◽  
pp. 63-72
Author(s):  
Ozkan Guner
Keyword(s):  
Mathematical Physics ◽  
Travelling Wave ◽  
Trigonometric Function ◽  
Hyperbolic Function ◽  
Travelling Wave Solutions ◽  
Long Wave ◽  
Regularized Long Wave Equation ◽  
Hyperbolic Function Solutions ◽  
Trigonometric Function Solutions ◽  
Fractional Differential

In this paper, solitary-wave ansatz and the (G?/G)-expansion methods have been used to obtain exact solutions of the fractional regularized long-wave (RLW) and coupled Nizhnik-Novikov-Veselov (NNV) equation. As a result, three types of exact analytical solutions such as rational function solutions, trigonometric function solutions, hyperbolic function solutions are formally derived these equations. Proposed methods are more powerful and can be applied to other fractional differential equations arising in mathematical physics.

scholarly journals New complex exact travelling wave solutions for the generalized-Zakharov equation with complex structures

2016 ◽  
Vol 6 (2) ◽  
pp. 141-150 ◽  
Author(s):  
Haci Mehmet Baskonus ◽  
Hasan Bulut
Keyword(s):  
Complex Structure ◽  
Three Dimensional ◽  
Expansion Method ◽  
Travelling Wave ◽  
Hyperbolic Function ◽  
Travelling Wave Solutions ◽  
Zakharov Equation ◽  
Wolfram Mathematica ◽  
Wave Solutions ◽  
Hyperbolic Function Solutions

In this paper, we apply the sine-Gordon expansion method which is one of the powerful methods to the generalized-Zakharov equation with complex structure. This algorithm yields new complex hyperbolic function solutions to the generalized-Zakharov equation with complex structure. Wolfram Mathematica 9 has been used throughout the paper for plotting two- and three-dimensional surface of travelling wave solutions obtained.

scholarly journals New Improvement of the Expansion Methods for Solving the Generalized Fitzhugh-Nagumo Equation with Time-Dependent Coefficients

2015 ◽  
Vol 2015 ◽  
pp. 1-35 ◽  
Author(s):  
Jalil Manafian ◽  
Mehrdad Lakestani
Keyword(s):  
Expansion Method ◽  
Travelling Wave ◽  
Trigonometric Function ◽  
Time Dependent ◽  
Hyperbolic Function ◽  
Travelling Wave Solutions ◽  
Partial Differential ◽  
Function Solution ◽  
Wave Solutions ◽  
Nagumo Equation

An improvement of the expansion methods, namely, the improved tan⁡Φξ/2-expansion method, for solving nonlinear second-order partial differential equation, is proposed. The implementation of the new approach is demonstrated by solving the generalized Fitzhugh-Nagumo equation with time-dependent coefficients. As a result, many new and more general exact travelling wave solutions are obtained including periodic function solutions, soliton-like solutions, and trigonometric function solutions. The exact particular solutions contain four types: hyperbolic function solution, trigonometric function solution, exponential solution, and rational solution. We obtained further solutions comparing this method with other methods. The results demonstrate that the new tan⁡Φξ/2-expansion method is more efficient than the Ansatz method and Tanh method applied by Triki and Wazwaz (2013). Recently, this method is developed for searching exact travelling wave solutions of nonlinear partial differential equations. Abundant exact travelling wave solutions including solitons, kink, and periodic and rational solutions have been found. These solutions might play an important role in engineering fields. It is shown that this method, with the help of symbolic computation, provides a straightforward and powerful mathematical tool for solving the nonlinear physics.

The Modified (G'/G)-Expansion Method for Nonlinear Evolution Equations

2011 ◽  
Vol 66 (1-2) ◽  
pp. 33-39 ◽  
Author(s):  
Sheng Zhang ◽  
Ying-Na Sun ◽  
Jin-Mei Ba ◽  
Ling Dong
Keyword(s):  
Evolution Equations ◽  
Nonlinear Evolution ◽  
Expansion Method ◽  
Travelling Wave ◽  
Nonlinear Evolution Equations ◽  
Hyperbolic Function ◽  
Mathematical Tool ◽  
Rational Solutions ◽  
Hyperbolic Function Solutions ◽  
Trigonometric Function Solutions

A modified (Gʹ/G)-expansion method is proposed to construct exact solutions of nonlinear evolution equations. To illustrate the validity and advantages of the method, the (3+1)-dimensional potential Yu-Toda-Sasa-Fukuyama (YTSF) equation is considered and more general travelling wave solutions are obtained. Some of the obtained solutions, namely hyperbolic function solutions, trigonometric function solutions, and rational solutions contain an explicit linear function of the variables in the considered equation. It is shown that the proposed method provides a more powerful mathematical tool for solving nonlinear evolution equations in mathematical physics.

scholarly journals Exact solutions of space–time fractional KdV–MKdV equation and Konopelchenko–Dubrovsky equation

Open Physics ◽  
2020 ◽  
Vol 18 (1) ◽  
pp. 871-880
Author(s):  
Bo Tang ◽  
Jiajia Tao ◽  
Shijun Chen ◽  
Junfeng Qu ◽  
Qian Wang ◽  
...  
Keyword(s):  
Exact Solutions ◽  
Expansion Method ◽  
Mkdv Equation ◽  
Space Time ◽  
Trigonometric Function ◽  
Hyperbolic Function ◽  
Rational Solutions ◽  
Hyperbolic Function Solutions ◽  
Trigonometric Function Solutions ◽  
Physical Features

Abstract In the present study, we deal with the space–time fractional KdV–MKdV equation and the space–time fractional Konopelchenko–Dubrovsky equation in the sense of the conformable fractional derivative. By means of the extend \left(\tfrac{G^{\prime} }{G}\right) -expansion method, many exact solutions are obtained, which include hyperbolic function solutions, trigonometric function solutions and rational solutions. The results show that the extend \left(\tfrac{G^{\prime} }{G}\right) -expansion method is an efficient technique for solving nonlinear fractional partial equations. We also provide some graphical representations to demonstrate the physical features of the obtained solutions.

scholarly journals Exact solutions with arbitrary functions of the (4+1)-dimensional fokas equation

2019 ◽  
Vol 23 (4) ◽  
pp. 2403-2411 ◽  
Author(s):  
Bo Xu ◽  
Sheng Zhang
Keyword(s):  
Exact Solutions ◽  
Elliptic Function ◽  
Expansion Method ◽  
Trigonometric Function ◽  
High Dimensional ◽  
Hyperbolic Function ◽  
Jacobi Elliptic Function ◽  
New Exact Solutions ◽  
Hyperbolic Function Solutions ◽  
Trigonometric Function Solutions

In this paper, the (4+1)-dimensional Fokas equation is solved by the generalized F-expansion method, and new exact solutions with arbitrary functions are obtained. The obtained solutions include Jacobi elliptic function solutions, hyperbolic function solutions and trigonometric function solutions. It is shown that the generalized F-expansion method can be used for constructing exact solutions with arbitrary functions of some other high dimensional partial differential equations in fluids.

scholarly journals The Investigation of Solutions to the Coupled Schrödinger-Boussinesq Equations

2013 ◽  
Vol 2013 ◽  
pp. 1-5 ◽  
Author(s):  
Xin Huang
Keyword(s):  
Rational Function ◽  
Symbolic Computation ◽  
Boussinesq Equations ◽  
Expansion Method ◽  
Trigonometric Function ◽  
Hyperbolic Function ◽  
Hyperbolic Function Solutions ◽  
Trigonometric Function Solutions

The (G′/G)-expansion method and the symbolic computation system Mathematica are employed to investigate the coupled Schrödinger-Boussinesq equations. The hyperbolic function solutions, trigonometric function solutions, and rational function solutions to the equations are obtained. The decaying properties of several solutions are analyzed.

scholarly journals Bifurcation and Solitary-Like Solutions for Compound KdV-Burgers-Type Equation

2015 ◽  
Vol 2015 ◽  
pp. 1-8
Author(s):  
Yin Li ◽  
Ruiying Wei ◽  
Guoming Jian
Keyword(s):  
Dynamical Systems ◽  
Type Equation ◽  
Trigonometric Function ◽  
Hyperbolic Function ◽  
Bifurcation Method ◽  
Dynamical Characteristics ◽  
Hyperbolic Function Solutions ◽  
Trigonometric Function Solutions ◽  
Ode Method ◽  
Parameter Condition

Firstly, based on the improved sub-ODE method and the bifurcation method of dynamical systems, we investigate the bifurcation of solitary waves in the compound KdV-Burgers-type equation. Secondly, numbers of solitary patterns solutions are given for each parameter condition and numerical simulations are used to display the dynamical characteristics. Finally, we obtain twelve solitary patterns solutions under some parameter conditions, such as the trigonometric function solutions and the hyperbolic function solutions.

scholarly journals A New Variable-Coefficient Riccati Subequation Method for Solving Nonlinear Lattice Equations

2013 ◽  
Vol 2013 ◽  
pp. 1-6 ◽  
Author(s):  
Fanwei Meng
Keyword(s):  
Variable Coefficient ◽  
Toda Lattice ◽  
Traveling Wave Solutions ◽  
Trigonometric Function ◽  
Hyperbolic Function ◽  
Lattice Equation ◽  
New Exact Solutions ◽  
Hyperbolic Function Solutions ◽  
Trigonometric Function Solutions ◽  
Toda Lattice Equation

We propose a new variable-coefficient Riccati subequation method to establish new exact solutions for nonlinear differential-difference equations. For illustrating the validity of this method, we apply it to the discrete (2 + 1)-dimensional Toda lattice equation. As a result, some new and generalized traveling wave solutions including hyperbolic function solutions, trigonometric function solutions, and rational function solutions are obtained.

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