The aim of this paper is to generate exact travelling wave solutions of the Harry-Dym equation through the methods of Adomian decomposition,He's variational iteration,direct integration,and power series.We show th...The aim of this paper is to generate exact travelling wave solutions of the Harry-Dym equation through the methods of Adomian decomposition,He's variational iteration,direct integration,and power series.We show that the two later methods axe more successful than the two former to obtain more solutions of the equation.展开更多
The hyperbolic function method for nonlinear wave equations ispresented. In support of a computer algebra system, many exact solitary wave solutions of a class of nonlinear wave equations are obtained via the method. ...The hyperbolic function method for nonlinear wave equations ispresented. In support of a computer algebra system, many exact solitary wave solutions of a class of nonlinear wave equations are obtained via the method. The method is based on the fact that the solitary wave solutions are essentially of a localized nature. Writing the solitary wave solutions of a nonlinear wave equation as the polynomials of hyperbolic functions, the nonlinear wave equation can be changed into a nonlinear system of algebraic equations. The system can be solved via Wu Elimination or Grbner base method. The exact solitary wave solutions of the nonlinear wave equation are obtained including many new exact solitary wave solutions.展开更多
An extended hyperbola function method is proposed to construct exact solitary wave solutions to nonlinear wave equation based upon a coupled Riccati equation. It is shown that more new solitary wave solutions can be f...An extended hyperbola function method is proposed to construct exact solitary wave solutions to nonlinear wave equation based upon a coupled Riccati equation. It is shown that more new solitary wave solutions can be found by this new method, which include kink-shaped soliton solutions, bell-shaped soliton solutions and new solitary wave.The new method can be applied to other nonlinear equations in mathematical physics.展开更多
The homogeneous balance method was improved and applied to two systems Of nonlinear evolution equations. As a result, several families of exact analytic solutions are derived by some new ansatzs. These solutions conta...The homogeneous balance method was improved and applied to two systems Of nonlinear evolution equations. As a result, several families of exact analytic solutions are derived by some new ansatzs. These solutions contain Wang's and Zhang's results and other new types of analytical solutions, such as rational fraction solutions and periodic solutions. The way can also be applied to solve more nonlinear partial differential equations.展开更多
In this paper, the Clarkson–Kruskal direct approach is employed to investigate the exact solutions of the2-dimensional rotational Euler equations for the incompressible fluid. The application of the method leads to a...In this paper, the Clarkson–Kruskal direct approach is employed to investigate the exact solutions of the2-dimensional rotational Euler equations for the incompressible fluid. The application of the method leads to a system of completely solvable ordinary differential equations. Several special cases are discussed and novel nonlinear exact solutions with respect to variables x and y are obtained. It is of interest to notice that the pressure p is obtained by the second kind of curvilinear integral and the coefficients of the nonlinear solutions are solitary wave type functions like tanh(kt/2)and sech(kt/2) due to the rotational parameter k = 0. Such phenomenon never appear in the classical Euler equations wherein the Coriolis force arising from the gravity and Earth's rotation is ignored. Finally, illustrative numerical figures are attached to show the behaviors that the exact solutions may exhibit.展开更多
Using direct algebraic method,exact solitary wave solutions are performed for a class of third order nonlinear dispersive disipative partial differential equations. These solutions are obtained under certain condition...Using direct algebraic method,exact solitary wave solutions are performed for a class of third order nonlinear dispersive disipative partial differential equations. These solutions are obtained under certain conditions for the relationship between the coefficients of the equation. The exact solitary waves of this class are rational functions of real exponentials of kink-type solutions.展开更多
The exact chirped soliton-like and quasi-periodic wave solutions of (2+1)-dimensional generalized nonlinearSchrodinger equation including linear and nonlinear gain (loss) with variable coefficients are obtained detail...The exact chirped soliton-like and quasi-periodic wave solutions of (2+1)-dimensional generalized nonlinearSchrodinger equation including linear and nonlinear gain (loss) with variable coefficients are obtained detailedly in thispaper.The form and the behavior of solutions are strongly affected by the modulation of both the dispersion coefhcientand the nonlinearity coefficient.In addition,self-similar soliton-like waves precisely piloted from our obtained solutionsby tailoring the dispersion and linear gain (loss).展开更多
In this paper,sub equation and (1=G’)-expansion methods are proposed to construct exact solutions of a nonlinear longitudinal wave equation(LWE)in a magneto-electro-elastic circular rod.The proposed methods have been...In this paper,sub equation and (1=G’)-expansion methods are proposed to construct exact solutions of a nonlinear longitudinal wave equation(LWE)in a magneto-electro-elastic circular rod.The proposed methods have been used to construct hyperbolic,rational,dark soliton and trigonometric solutions of the LWE in the magnetoelectro-elastic circular rod.Arbitrary values are given to the parameters in the solutions obtained.3D,2D and contour graphs are presented with the help of a computer package program.Solutions attained by symbolic calculations revealed that these methods are effective,reliable and simple mathematical tool for finding solutions of nonlinear evolution equations arising in physics and nonlinear dynamics.展开更多
The present paper studies the unstable nonlinear Schr¨odinger equations, describing the time evolution of disturbances in marginally stable or unstable media. More precisely, the unstable nonlinear Schr¨odin...The present paper studies the unstable nonlinear Schr¨odinger equations, describing the time evolution of disturbances in marginally stable or unstable media. More precisely, the unstable nonlinear Schr¨odinger equation and its modified form are analytically solved using two efficient distinct techniques, known as the modified Kudraysov method and the sine-Gordon expansion approach. As a result, a wide range of new exact traveling wave solutions for the unstable nonlinear Schr¨odinger equation and its modified form are formally obtained.展开更多
The solitary wave solutions of the combined KdV-mKdV-Burgers equation and the Kolmogorov-Petrovskii-Piskunov equation are obtained by means of the direct algebra method, which can be generalized to deal with high dime...The solitary wave solutions of the combined KdV-mKdV-Burgers equation and the Kolmogorov-Petrovskii-Piskunov equation are obtained by means of the direct algebra method, which can be generalized to deal with high dimensional nonlinear evolution equations.展开更多
This paper reflects the execution of a reliable technique which we proposed as a new method called the double auxiliary equations method for constructing new traveling wave solutions of nonlinear fractional differenti...This paper reflects the execution of a reliable technique which we proposed as a new method called the double auxiliary equations method for constructing new traveling wave solutions of nonlinear fractional differential equation.The proposed scheme has been successfully applied on two very important evolution equations,the space-time fractional differential equation governing wave propagation in low-pass electrical transmission lines equation and the time fractional Burger’s equation.The obtained results show that the proposed method is more powerful,promising and convenient for solving nonlinear fractional differential equations(NFPDEs).To our knowledge,the solutions obtained by the proposed method have not been reported in former literature.展开更多
In this article, the fractional derivatives in the sense of the modified Riemann-Liouville derivatives together with the modified simple equation method and the multiple exp-function method are employed for constructi...In this article, the fractional derivatives in the sense of the modified Riemann-Liouville derivatives together with the modified simple equation method and the multiple exp-function method are employed for constructing the exact solutions and the solitary wave solutions for the nonlinear time fractional Sharma-Tasso- Olver equation. With help of Maple, we can get exact explicit l-wave, 2-wave and 3-wave solutions, which include l-soliton, 2-soliton and 3-soliton type solutions if we use the multiple exp-function method while we can get only exact explicit l-wave solution including l-soliton type solution if we use the modified simple equation method. Two cases with specific values of the involved parameters are plotted for each 2-wave and 3-wave solutions.展开更多
文摘The aim of this paper is to generate exact travelling wave solutions of the Harry-Dym equation through the methods of Adomian decomposition,He's variational iteration,direct integration,and power series.We show that the two later methods axe more successful than the two former to obtain more solutions of the equation.
基金This work was supported by the National 973 Project (Grant No. G1998030600) Post-doctoral Foundation .
文摘The hyperbolic function method for nonlinear wave equations ispresented. In support of a computer algebra system, many exact solitary wave solutions of a class of nonlinear wave equations are obtained via the method. The method is based on the fact that the solitary wave solutions are essentially of a localized nature. Writing the solitary wave solutions of a nonlinear wave equation as the polynomials of hyperbolic functions, the nonlinear wave equation can be changed into a nonlinear system of algebraic equations. The system can be solved via Wu Elimination or Grbner base method. The exact solitary wave solutions of the nonlinear wave equation are obtained including many new exact solitary wave solutions.
文摘An extended hyperbola function method is proposed to construct exact solitary wave solutions to nonlinear wave equation based upon a coupled Riccati equation. It is shown that more new solitary wave solutions can be found by this new method, which include kink-shaped soliton solutions, bell-shaped soliton solutions and new solitary wave.The new method can be applied to other nonlinear equations in mathematical physics.
文摘The homogeneous balance method was improved and applied to two systems Of nonlinear evolution equations. As a result, several families of exact analytic solutions are derived by some new ansatzs. These solutions contain Wang's and Zhang's results and other new types of analytical solutions, such as rational fraction solutions and periodic solutions. The way can also be applied to solve more nonlinear partial differential equations.
基金Supported by the National Natural Science Foundation of China under Grant No.11301269Jiangsu Provincial Natural Science Foundation of China under Grant No.BK20130665+2 种基金the Fundamental Research Funds KJ2013036 for the Central UniversitiesStudent Research Training under Grant No.1423A02 of Nanjing Agricultural Universitythe Research Grant RG21/2013-2014R from the Hong Kong Institute of Education
文摘In this paper, the Clarkson–Kruskal direct approach is employed to investigate the exact solutions of the2-dimensional rotational Euler equations for the incompressible fluid. The application of the method leads to a system of completely solvable ordinary differential equations. Several special cases are discussed and novel nonlinear exact solutions with respect to variables x and y are obtained. It is of interest to notice that the pressure p is obtained by the second kind of curvilinear integral and the coefficients of the nonlinear solutions are solitary wave type functions like tanh(kt/2)and sech(kt/2) due to the rotational parameter k = 0. Such phenomenon never appear in the classical Euler equations wherein the Coriolis force arising from the gravity and Earth's rotation is ignored. Finally, illustrative numerical figures are attached to show the behaviors that the exact solutions may exhibit.
文摘Using direct algebraic method,exact solitary wave solutions are performed for a class of third order nonlinear dispersive disipative partial differential equations. These solutions are obtained under certain conditions for the relationship between the coefficients of the equation. The exact solitary waves of this class are rational functions of real exponentials of kink-type solutions.
基金Supported by the National Natural Science Foundation of China under Grant No.11072219the Zhejiang Provincial Natural Science Foundation under Grant No.Y1100099
文摘The exact chirped soliton-like and quasi-periodic wave solutions of (2+1)-dimensional generalized nonlinearSchrodinger equation including linear and nonlinear gain (loss) with variable coefficients are obtained detailedly in thispaper.The form and the behavior of solutions are strongly affected by the modulation of both the dispersion coefhcientand the nonlinearity coefficient.In addition,self-similar soliton-like waves precisely piloted from our obtained solutionsby tailoring the dispersion and linear gain (loss).
文摘In this paper,sub equation and (1=G’)-expansion methods are proposed to construct exact solutions of a nonlinear longitudinal wave equation(LWE)in a magneto-electro-elastic circular rod.The proposed methods have been used to construct hyperbolic,rational,dark soliton and trigonometric solutions of the LWE in the magnetoelectro-elastic circular rod.Arbitrary values are given to the parameters in the solutions obtained.3D,2D and contour graphs are presented with the help of a computer package program.Solutions attained by symbolic calculations revealed that these methods are effective,reliable and simple mathematical tool for finding solutions of nonlinear evolution equations arising in physics and nonlinear dynamics.
文摘The present paper studies the unstable nonlinear Schr¨odinger equations, describing the time evolution of disturbances in marginally stable or unstable media. More precisely, the unstable nonlinear Schr¨odinger equation and its modified form are analytically solved using two efficient distinct techniques, known as the modified Kudraysov method and the sine-Gordon expansion approach. As a result, a wide range of new exact traveling wave solutions for the unstable nonlinear Schr¨odinger equation and its modified form are formally obtained.
基金This work was supported by NSF of Guangdong Proovince (20010026).
文摘The solitary wave solutions of the combined KdV-mKdV-Burgers equation and the Kolmogorov-Petrovskii-Piskunov equation are obtained by means of the direct algebra method, which can be generalized to deal with high dimensional nonlinear evolution equations.
文摘This paper reflects the execution of a reliable technique which we proposed as a new method called the double auxiliary equations method for constructing new traveling wave solutions of nonlinear fractional differential equation.The proposed scheme has been successfully applied on two very important evolution equations,the space-time fractional differential equation governing wave propagation in low-pass electrical transmission lines equation and the time fractional Burger’s equation.The obtained results show that the proposed method is more powerful,promising and convenient for solving nonlinear fractional differential equations(NFPDEs).To our knowledge,the solutions obtained by the proposed method have not been reported in former literature.
文摘In this article, the fractional derivatives in the sense of the modified Riemann-Liouville derivatives together with the modified simple equation method and the multiple exp-function method are employed for constructing the exact solutions and the solitary wave solutions for the nonlinear time fractional Sharma-Tasso- Olver equation. With help of Maple, we can get exact explicit l-wave, 2-wave and 3-wave solutions, which include l-soliton, 2-soliton and 3-soliton type solutions if we use the multiple exp-function method while we can get only exact explicit l-wave solution including l-soliton type solution if we use the modified simple equation method. Two cases with specific values of the involved parameters are plotted for each 2-wave and 3-wave solutions.
