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.

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.

scholarly journals On a direct construction of inverse scattering problems for integrable nonlinear evolution equations in the two-spatial dimension

2004 ◽  
Vol 2004 (58) ◽  
pp. 3117-3128
Author(s):  
H. H. Chen ◽  
J. E. Lin
Keyword(s):  
Inverse Scattering ◽  
Evolution Equations ◽  
Nonlinear Evolution ◽  
Spatial Dimension ◽  
Nonlinear Evolution Equations ◽  
Scattering Problems ◽  
Direct Construction ◽  
Inverse Scattering Problems ◽  
Petviashvili Equation ◽  
The One

We present a method to construct inverse scattering problems for integrable nonlinear evolution equations in the two-spatial dimension. The temporal component is the adjoint of the linearized equation and the spatial component is a partial differential equation with respect to the spatial variables. Although this idea has been known for the one-spatial dimension for some time, it is the first time that this method is presented for the case of the higher-spatial dimension. We present this method in detail for the Veselov-Novikov equation and the Kadomtsev-Petviashvili equation.

Application of Extended Transformed Rational Function Method to Some (3+1) Dimensional Nonlinear Evolution Equations

2018 ◽  
pp. 433-437
Keyword(s):  
Rational Function ◽  
Function Method ◽  
Evolution Equations ◽  
Nonlinear Evolution ◽  
Kdv Equation ◽  
Mapping Method ◽  
Nonlinear Evolution Equations ◽  
Petviashvili Equation ◽  
Complexiton Solutions ◽  
Transformed Rational Function Method

The transformed rational function method can be considered as unification of the tanh type methods, the homogeneous balance method, the mapping method, the exp-function method and the F-expansion type methods. In this paper, we present complexiton solutions of (3+1) dimensional Korteweg-de Vries (KdV) equation and a new (3+1) dimensional generalized Kadomtsev-Petviashvili equation by using extended transformed rational function method which provides very useful and effective way to obtain complexiton solutions of nonlinear evolution equations.

NONLINEAR EVOLUTION EQUATIONS AND THE PAINLEVÉ TEST

1992 ◽  
Vol 07 (08) ◽  
pp. 1669-1683 ◽  
Author(s):  
W.-H. STEEB ◽  
N. EULER
Keyword(s):  
Evolution Equations ◽  
Gordon Equation ◽  
Nonlinear Evolution ◽  
Energy Eigenvalue ◽  
Nonlinear Evolution Equations ◽  
The Self ◽  
Painlevé Test ◽  
Yang Mills ◽  
Petviashvili Equation ◽  
Klein Gordon

A survey is given of new results of the Painlevé test and nonlinear evolution equations where ordinary- and partial-differential equations are considered. We study the semiclassical Jaynes-Cumming model, the energy-eigenvalue-level-motion equation, the Kadomtsev-Petviashvili equation, the nonlinear Klein-Gordon equation and the self-dual Yang-Mills equation.

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.

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