Solitons and rational solutions of nonlinear evolution equations

1978 ◽  
Vol 19 (10) ◽  
pp. 2180 ◽  
Author(s):  
M. J. Ablowitz ◽  
J. Satsuma
Keyword(s):  
Evolution Equations ◽  
Nonlinear Evolution ◽  
Nonlinear Evolution Equations ◽  
Rational Solutions

Abundant New Multiple Soliton-like Solutions and Rational Solutions of the (2+1)-Dimensional Broer-Kaup Equation

2001 ◽  
Vol 56 (12) ◽  
pp. 816-824 ◽  
Author(s):  
Zhenya Yan
Keyword(s):  
Evolution Equations ◽  
Nonlinear Evolution ◽  
Bäcklund Transformation ◽  
Soliton Solutions ◽  
Nonlinear Evolution Equations ◽  
Heat Equations ◽  
Backlund Transformation ◽  
Rational Solutions ◽  
Homogeneous Balance Method ◽  
Homogeneous Balance

Abstract In this paper we firstly improve the homogeneous balance method due to Wang, which was only used to obtain single soliton solutions of nonlinear evolution equations, and apply it to (2 + 1)-dimensional Broer-Kaup (BK) equations such that a Backlund transformation is found again. Considering further the obtained Backlund transformation, the relations are deduced among BK equations, well-known Burgers equations and linear heat equations. Finally, abundant multiple soliton-like solutions and infinite rational solutions are obtained from the relations.

scholarly journals Multi-soliton rational solutions for some nonlinear evolution equations

Open Physics ◽  
2016 ◽  
Vol 14 (1) ◽  
pp. 26-36 ◽  
Author(s):  
Mohamed S. Osman
Keyword(s):  
Evolution Equations ◽  
Nonlinear Evolution ◽  
Inverse Scattering Method ◽  
Nonlinear Evolution Equations ◽  
Scattering Method ◽  
Mathematical Tool ◽  
Rational Solutions ◽  
Generalized Unified Method ◽  
The Generalized Unified Method ◽  
Unified Method

AbstractThe Korteweg-de Vries equation (KdV) and the (2+ 1)-dimensional Nizhnik-Novikov-Veselov system (NNV) are presented. Multi-soliton rational solutions of these equations are obtained via the generalized unified method. The analysis emphasizes the power of this method and its capability of handling completely (or partially) integrable equations. Compared with Hirota’s method and the inverse scattering method, the proposed method gives more general exact multi-wave solutions without much additional effort. The results show that, by virtue of symbolic computation, the generalized unified method may provide us with a straightforward and effective mathematical tool for seeking multi-soliton rational solutions for solving many nonlinear evolution equations arising in different branches of sciences.

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 Weakly Coupled B-Type Kadomtsev-Petviashvili Equation: Lump and Rational Solutions

2020 ◽  
Vol 2020 ◽  
pp. 1-8
Author(s):  
Na Xiong ◽  
Wen-Tao Li ◽  
Biao Li ◽  
Zine El Abiddine Fellah
Keyword(s):  
Numerical Analysis ◽  
Evolution Equations ◽  
Nonlinear Evolution ◽  
Positive Semidefinite ◽  
Nonlinear Evolution Equations ◽  
Rational Solutions ◽  
Weakly Coupled ◽  
Petviashvili Equation ◽  
Trajectory Equation ◽  
Semidefinite Matrix

Through the method of Z N -KP hierarchy, we propose a new ( 3 + 1 )-dimensional weakly coupled B-KP equation. Based on the bilinear form, we obtain the lump and rational solutions to the dimensionally reduced cases by constructing a symmetric positive semidefinite matrix. Then, we do numerical analysis on the rational solutions and fit the trajectory equation of the crest. Furthermore, we verify the accuracy of the trajectory equation by numerical analysis. This method of solving the lump and rational solutions can also be applied to other nonlinear evolution equations.

EXACT TRAVELING WAVE SOLUTIONS FOR SOME NONLINEAR EVOLUTION EQUATIONS WITH NONLINEAR TERMS OF ANY ORDER

2003 ◽  
Vol 14 (01) ◽  
pp. 99-112 ◽  
Author(s):  
YONG CHEN ◽  
BIAO LI ◽  
HONG-QING ZHANG
Keyword(s):  
Solitary Wave ◽  
Traveling Wave ◽  
Evolution Equations ◽  
Nonlinear Evolution ◽  
Traveling Wave Solutions ◽  
Nonlinear Evolution Equations ◽  
Solitary Wave Solutions ◽  
Rational Solutions ◽  
Exact Traveling Wave Solutions ◽  
Wave Solutions

In this paper, we improved the tanh method by means of a proper transformation and general ansätz. Using the improved method, with the aid of Mathematica™, we consider some nonlinear evolution equations with nonlinear terms of any order. As a result, rich explicit exact traveling wave solutions for these equations, which contain kink profile solitary wave solutions, bell profile solitary wave solutions, rational solutions, periodic solutions, and combined formal solutions, are obtained.

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