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Double-Parameter Solutions of Projective Riccati Equations and Their Applications

Communications in Theoretical Physics 2007年第1期47卷 1-9页

作者： WANG Ming-Liang LI Er-Qiang LI Xiang-Zheng Science College Henan University of Science and Technology Luoyang 471003 China Department of Mathematics Lanzhou University Lanzhou 730000 China

The purpose of the present paper is twofold. First, the projective Riccati equations （PREs for short） are resolved by means of a linearized theorem, which was known in the literature. Based on the signs and values of coeffcients of PREs, the solutions with two arbitrary parameters of PREs can be expressed by the hyperbolic functions, the trigonometric functions, and the rational functions respectively, at the same time the relation between the components of each solution to PREs is also implemented. Second, more new travelling wave solutions for some nonlinear PDEs, such as the Burgers equation, the mKdV equation, the NLS^＋ equation, new Hamilton amplitude equation, and so on, are obtained by using Sub-ODE method, in which PREs are taken as the Sub-ODEs. The key idea of this method is that the travelling wave solutions of nonlinear PDE can be expressed by a polynomial in two variables, which are the components of each solution to PREs, provided that the homogeneous balance between the higher order derivatives and nonlinear terms in the equation is considered.

关键词： projective Riccati equations linearized theorem homogeneous balance Sub-ODE method travelling wave solutions nonlinear PDE

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Solitary Wave Solutions to the ZKBBM Equation and the KPBBM Equation via the Modified Simple Equation Method

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Journal of Partial Differential Equations 2016年第2期29卷 143-160页

作者： AKTER J. AKBAR M. Ali Department of Natural Science Varendra University Bangladesh. Department of Applied Mathematics University of Rajshahi Bangladesh.

In this article, the modified simple equation method （MSE） is used to acquire exact solutions to nonlinear evolution equations （NLEEs） namely the Zakharov- Kuznetsov Benjamin-Bona-Mahony equation and the Kadomtsov-Petviashvilli Benjamin-Bona-Mahony equation which have widespread usage in modern science. The MSE method is ascending and useful mathematical tool for constructing exact travel- ing wave solutions to NLEEs in the field of science and engineering. By means of this method we attained some significant solutions with free parameters and for special values of these parameters, we found some soliton solutions derived from the exact solutions. The solutions obtained in this article have been shown graphically and also discussed physically.

关键词： Modified simple equation method nonlinear evolution equations homogeneous balance soliton solutions Zakharov-Kuznetsov Benjamin-Bona-Mahony equation Kadomtsov- Petviashvilli Benjamin-Bona-Mahony equation.

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Exact Travelling Wave Solutions for a Combined KdV and Schwarzian KdV Equation

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Chinese Quarterly Journal of Mathematics 2012年第3期27卷 371-374页

作者： LI Ling-xiao LI Er-qiang College of Mathematics and Statistics Henan University of Science and Technology Luoyang 471003China

In this paper the ( G'/G )-expansion method is used to find exact travelling wave solutions for a combined KdV and Schwarzian KdV equation. As a result, multiple travelling wave solutions with arbitrary parameters are obtained, which are expressed by hyperbolic functions, trigonometric functions and rational functions. When the parameters are taken as special values, the solitary waves are derived from the travelling waves. The (G'/G)-expansion method presents a wider applicability for handling nonlinear wave equations.

关键词： the (G’/G)-expansion method travelling wave solutions homogeneous balance the combined KdV and Schwarzian KdV equation

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