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Hirota bilinear equations with linear subspaces of hyperbolic and trigonometric function solutions

APPLIED MATHEMATICS AND COMPUTATION 2013年 220卷 226-234页

作者： Zheng, Hong-Chan Ma, Wen-Xiu Gu, Xiang Northwestern Polytech Univ Dept Appl Math Xian 710072 Peoples R China Univ S Florida Dept Math & Stat Tampa FL 33620 USA

Linear superposition principles of hyperbolic and trigonometric function solutions are analyzed for Hirota bilinear equations, with an aim to construct a specific sub-class of N-soliton solutions formulated by linear combinations of hyperbolic and trigonometric functions. An algorithm using weights is discussed and a few illustrative application examples are presented. (C) 2013 Elsevier Inc. All rights reserved.

关键词： Hirota bilinear form Linear subspace of solutions Hyperbolic function solution trigonometric function solution

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An improved generalized F-expansion method and its application to the (2+1)-dimensional KdV equations

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COMMUNICATIONS IN NONLINEAR SCIENCE AND NUMERICAL SIMULATION 2008年第7期13卷 1294-1301页

作者： Zhang, Sheng Xia, Tiecheng Bohai Univ Dept Math Jinzhou 121000 Peoples R China Shanghai Univ Dept Math Shanghai 200444 Peoples R China

An improved generalized F-expansion method is proposed to seek exact solutions of nonlinear partial differential equations. With the aid of symbolic computation, we choose the (2 + 1)-dimensional KdV equations to illustrate the validity and advantages of the proposed method. Many new and more general non-travelling wave solutions are obtained, including single and combined non-degenerate Jacobi elliptic function solutions, soliton-like solutions, trigonometric function solutions, each of which contains two arbitrary functions. (C) 2007 Elsevier B.V. All rights reserved.

关键词： F-expansion method Jacobi elliptic function soliton-like solution trigonometric function solution

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EXACT solutionS WITH EXTERNAL LINEAR functionS FOR THE POTENTIAL YU-TODA-SASA-FUKUYAMA EQUATION

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THERMAL SCIENCE 2018年第4期22卷 1621-1628页

作者： Zhang, Sheng Zong, Qianan Bohai Univ Sch Math & Phys Jinzhou Peoples R China

Constructing exact solutions of non-linear PDE is of both theoretical and practical values. In this paper, a modified F-expansion method is proposed to construct exact solutions of non-linear PDE. To illustrate the validity and advantages of the proposed method, a (3+1)-D potential Yu-Toda-Sasa-Fukuyama equation is considered and more general exact solutions with external linear functions are obtained including Jacobi elliptic function solutions, hyperbolic function solutions, and trigonometric function solutions. It is shown that the original F-expansion method can not construct exact solutions of the potential Yu-Toda-Sasa-Fukuyama equation but the modified method is valid. The modified F-expansion method can lead to such exact solutions with external linear functions which possess a remarkable dynamical property, which is the solitary wave does not propagate in the horizontal direction as the traditional waves. The modified F-expansion method can be used for exactly solving some other non-linear PDE.

关键词： potential Yu-Toda-Sasa-Fukuyama equation modified F-expansion method Jacobi elliptic function solution hyperbolic function solution trigonometric function solution

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Application of tan(φ/2)-expansion method for solving the Biswas-Milovic equation for Kerr law nonlinearity

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OPTIK 2016年第4期127卷 2040-2054页

作者： Manafian, Jalil Lakestani, Mehrdad Univ Tabriz Fac Math Sci Dept Appl Math Tabriz Iran

In this paper, the improved tan(Phi(xi)/2)-expansion method is proposed to seek more general exact solutions of the new partial differential equation. Being concise and straightforward, this method is applied to the Biswas-Milovic equation (BME). The exact particular solutions containing four types hyperbolic function solution, trigonometric function solution, exponential solution and rational solution. We obtained the further solutions comparing with other methods as [13]. Recently this method is developed for searching exact traveling wave solutions of nonlinear partial differential equations. Abundant exact traveling wave solutions including solitons, kink, periodic and rational solutions have been found. These solutions might play important role in engineering and physics fields. It is shown that this method, with the help of symbolic computation, provide a straightforward and powerful mathematical tool for solving problems in nonlinear optic. (C) 2015 Elsevier GmbH. All rights reserved.

关键词： Improved tan(Phi)(xi)/2)-expansion method Biswas-Milovic equation Hyperbolic function solution trigonometric function solution Exponential solution and rational solution

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The correct traveling wave solutions for the high-order dispersive nonlinear Schrodinger equation

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APPLIED MATHEMATICS AND COMPUTATION 2010年第5期216卷 1583-1591页

作者： Shi, Yeqiong Dai, Zhengde Li, Donglong Guangxi Univ Technol Dept Informat & Comp Sci Guangxi 545006 Liuzhou Peoples R China Yunnan Univ Sch Math & Stat Kunming 650091 Peoples R China

The high-order dispersive nonlinear Schrodinger equation is considered. The exact solutions were obtained by Zhang et al. [J.L. Zhang, M. L. Wang, X.Z. Li, Phys. Lett. A 357 (2006) 188-195] are analyzed. We can demonstrate that some solutions do not satisfy this equation. To obtain the correct solutions, the F-expansion method is applied to solve it. (C) 2010 Elsevier Inc. All rights reserved.

关键词： F-expansion method Higher order dispersive nonlinear Schrodinger equation Jacobi elliptic function Soliton-like solution trigonometric function solution

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Optical soliton solutions for Schrodinger type nonlinear evolution equations by the tan(Φ(ξ)/2)-expansion method

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OPTIK 2016年第10期127卷 4222-4245页

作者： Manafian, Jalil Islamic Azad Univ Ilkhchi Branch Young Researchers & Elite Club Ilkhchi Iran

We analytically study the Schrodinger type nonlinear evolution equations by improved tan(Phi(xi)/2)-expansion method. Explicit solutions are derived, which include optical bright solitons, dark solitons, singular solutions and singular triangular periodic solution. The exact particular solutions containing four types hyperbolic function solution, trigonometric function solution, exponential solution and rational solution. We obtained the further solutions comparing with other methods. Recently this method is developed for searching exact travelling wave solutions of nonlinear partial differential equations. It is shown that this method, with the help of symbolic computation, provide a straightforward and powerful mathematical tool for solving problems in nonlinear optic. (C) 2016 Elsevier GmbH. All rights reserved.

关键词： Improved tan(Phi(xi)/2)-expansion method Schrodinger equation Hyperbolic function solution trigonometric function solution Exponential solution and rational solution

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An improved generalized F-expansion method and its applications

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JOURNAL OF COMPUTATIONAL AND APPLIED MATHEMATICS 2008年第1期214卷 202-208页

作者： Abdou, M. A. Mansoura Univ Fac Sci Dept Phys Theoret Res Grp Mansoura 35516 Egypt King Kahlid Univ Dept Phys Fac Educ Girls Bisha Saudi Arabia

An improved generalized F-expansion method is proposed to seek exact solutions of nonlinear evolution equations. With the aid of symbolic computation, we choose the generalized nonlinear Schrodinger equation with a source to illustrate the validity and advantages of the proposed method. Many new and more general non-travelling wave solutions are obtained, including single and combined non-degenerate Jacobi elliptic function solutions, solitons-like solutions. trigonometric function solutions and rational solutions. The method is straightforward and concise, and it can also be applied to other nonlinear evolution equations in physics. (C) 2007 Elsevier B.V. All rights reserved.

关键词： F-expansion method Jacobi elliptic function soliton-like solution trigonometric function solution

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Variable separation method for a nonlinear time fractional partial differential equation with forcing term

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JOURNAL OF COMPUTATIONAL AND APPLIED MATHEMATICS 2018年 339卷 297-305页

作者： Zhang, Sheng Hong, Siyu Bohai Univ Sch Math & Phys Jinzhou 121013 Peoples R China

In this paper, a variable-coefficient nonlinear time fractional partial differential equation (PDE) with initial and boundary conditions is solved by using the variable separation method. As a result, some new explicit and exact solutions of the time fractional PDE are obtained including Airy function solution, hyperbolic function solution, trigonometric function solution and rational solution. It is shown that the variable separation method can provide a useful mathematical tool for solving some other nonlinear time fractional PDEs in science and engineering. (C) 2017 Elsevier B.V. All rights reserved.

关键词： Variable separation method Time fractional PDE Airy function solution Hyperbolic function solution trigonometric function solution Rational solution

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On exact solutions of modified KdV-ZK equation

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ALEXANDRIA ENGINEERING JOURNAL 2016年第4期55卷 3253-3265页

作者： Mohyud-Din, Syed Tauseef Irshad, Amna HITEC Univ Fac Sci Dept Math Taxila Cantt Pakistan

In this work, we established some exact particular solutions with parameters for Modified KdV-ZK Equation. The improved tan(phi(xi)/2) -expansion method is introduced to construct exact particular solutions of nonlinear evolution equations. The exact particular solutions contain four types: hyperbolic function solution, trigonometric function solution, exponential solution, and rational solution. The method appears to be effective mathematical tool for solving nonlinear evolution equations (NLEEs) in applied mathematics and engineering. (C) 2016 Faculty of Engineering, Alexandria University. Production and hosting by Elsevier B.V.

关键词： Improved tan(phi(xi)/2)-expansion method Hyperbolic function solution trigonometric function solution Rational solution Modified KdV-ZK equation

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The new complex rational function prototype structures for the nonlinear Schrodinger-inviscid burgers system

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MATHEMATICAL METHODS IN THE APPLIED SCIENCES 2018年第16期41卷 6312-6325页

作者： Ray, Santanu Saha Natl Inst Technol Dept Math Rourkela 769008 India

In this paper, a system representing the coupling between the nonlinear Schrodinger equation and the inviscid burgers equation in modeling the interactions between short and long waves in fluids has been investigated. The new algorithms, like improved tan(phi(xi)/2)-expansion method and improved Bernoulli subequation function method, have been proposed. The proposed methods have been discussed comprehensively in this article. Using these methods, some new prototype for exact solutions such as complex exponential, complex hyperbolic, and complex trigonometric function solutions have been obtained for nonlinear Schrodinger-inviscid burgers system. Through the present analysis, it has been established that, with the help of symbolic computation, these methods provide a straightforward and powerful mathematical tool for solving nonlinear evolution equations.

关键词： complex function solution exponential function solution hyperbolic function solution improved Bernoulli subequation function method improved content style-type=mathematics tan(phi() 2)-content>-expansion method nonlinear Schrodinger equation rational function solution Schrodinger-inviscid burgers system trigonometric function solution

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