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.

Exact Solutions for Coupled mKdV Equations by a New Symbolic Computation Method

2010 ◽  
Vol 20-23 ◽  
pp. 184-189 ◽  
Author(s):  
Bang Qing Li ◽  
Yu Lan Ma
Keyword(s):  
Exact Solutions ◽  
Symbolic Computation ◽  
Evolution Equations ◽  
Expansion Method ◽  
Nonlinear Evolution Equations ◽  
Hyperbolic Function ◽  
Computation Method ◽  
New Exact Solutions ◽  
Hyperbolic Function Solutions ◽  
Trigonometric Function Solutions

By introducing (G′/G)-expansion method and symbolic computation software MAPLE, two types of new exact solutions are constructed for coupled mKdV equations. The solutions included trigonometric function solutions and hyperbolic function solutions. The procedure is concise and straightforward, and the method is also helpful to find exact solutions for other nonlinear evolution equations.

A series of complexiton soliton solutions for nonlinear Jaulent—Miodek PDEs using the Riccati equations method

2011 ◽  
Vol 141 (5) ◽  
pp. 1001-1015 ◽  
Author(s):  
E. M. E. Zayed ◽  
Khaled A. Gepreel
Keyword(s):  
Rational Function ◽  
Symbolic Computation ◽  
Soliton Solutions ◽  
Expansion Method ◽  
Riccati Equations ◽  
Hyperbolic Function ◽  
Periodic Functions ◽  
Hyperbolic Functions ◽  
Hyperbolic Function Solutions ◽  
Triangular Function

We use the extended multiple Riccati equations expansion method to construct a series of double-soliton-like solutions, double triangular function solutions and complexiton soliton solutions for nonlinear Jaulent-Miodek PDEs. With the help of symbolic computation using Maple and Mathematica, we obtain many new types of complexiton soliton solutions, i.e. various combinations of trigonometric periodic functions and hyperbolic function solutions, various combinations of trigonometric periodic functions and rational function solutions, and various combinations of hyperbolic functions and rational function solutions.

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 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.

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.

A Class of Exact Solutions of (3+1)-Dimensional Generalized B-Type Kadomtsev–Petviashvili Equation

2017 ◽  
Vol 18 (2) ◽  
pp. 137-143 ◽  
Author(s):  
Shuang Liu ◽  
Yao Ding ◽  
Jian-Guo Liu
Keyword(s):  
Exact Solutions ◽  
Symbolic Computation ◽  
Traveling Wave ◽  
Expansion Method ◽  
Trigonometric Function ◽  
Hyperbolic Function ◽  
New Exact Solutions ◽  
Petviashvili Equation

AbstractBy employing the generalized$(G'/G)$-expansion method and symbolic computation, we obtain new exact solutions of the (3 + 1) dimensional generalized B-type Kadomtsev–Petviashvili equation, which include the traveling wave exact solutions and the non-traveling wave exact solutions showed by the hyperbolic function and the trigonometric function. Meanwhile, some interesting physics structure are discussed.

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.

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 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.

scholarly journals The Improvedexp⁡-Φξ-Expansion Method and New Exact Solutions of Nonlinear Evolution Equations in Mathematical Physics

2019 ◽  
Vol 2019 ◽  
pp. 1-8 ◽  
Author(s):  
Guiying Chen ◽  
Xiangpeng Xin ◽  
Hanze Liu
Keyword(s):  
Evolution Equations ◽  
Nonlinear Evolution ◽  
Traveling Wave Solutions ◽  
Expansion Method ◽  
Nonlinear Evolution Equations ◽  
Hyperbolic Function ◽  
Jacobi Elliptic Function ◽  
New Exact Solutions ◽  
Hyperbolic Function Solutions ◽  
Trigonometric Function Solutions

Theexp(-Φ(ξ))-expansion method is improved by presenting a new auxiliary ordinary differential equation forΦ(ξ). By using this method, new exact traveling wave solutions of two important nonlinear evolution equations, i.e., the ill-posed Boussinesq equation and the unstable nonlinear Schrödinger equation, are constructed. The obtained solutions contain Jacobi elliptic function solutions which can be degenerated to the hyperbolic function solutions and the trigonometric function solutions. The present method is very concise and effective and can be applied to other types of nonlinear evolution equations.

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