Truncated Painlevé Expansion with Symbolic Computation for a General Kadomtsev-Petviashvili Equation with Variable Coefficients

1996 ◽  
Vol 51 (3) ◽  
pp. 175-178
Author(s):  
Bo Tian ◽  
Yi-Tian Gao
Keyword(s):  
Symbolic Computation ◽  
Variable Coefficient ◽  
Bäcklund Transformation ◽  
Variable Coefficients ◽  
Explicit Solutions ◽  
Current Interest ◽  
Backlund Transformation ◽  
Petviashvili Equation ◽  
Mathematical Sciences ◽  
Truncated Painlevé Expansion

Able to realistically model various physical situations, the variable-coefficient generalizations of the celebrated Kadmotsev-Petviashvili equation are of current interest in physical and mathematical sciences. In this paper, we make use of both the truncated Painleve expansion and symbolic computation to obtain an auto-Bäcklund transformation and certain soliton-typed explicit solutions for a general Kadomtsev-Petviashvili equation with variable coefficients.

Incompressible-Fluid Symbolic Computation and Bäcklund Transformation: (3+1)-Dimensional Variable-Coefficient Boiti–Leon–Manna–Pempinelli Model

2015 ◽  
Vol 70 (1) ◽  
pp. 59-61 ◽  
Author(s):  
Xin-Yi Gao
Keyword(s):  
Shock Wave ◽  
Incompressible Fluid ◽  
Symbolic Computation ◽  
Variable Coefficient ◽  
Bäcklund Transformation ◽  
Incompressible Fluids ◽  
Current Interest ◽  
Backlund Transformation ◽  
Wave Type ◽  
Dimensional Variable

AbstractIncompressible fluids are of current interest. Considering a (3+1)-dimensional variable-coefficient Boiti–Leon–Manna–Pempinelli model for an incompressible fluid, we perform symbolic computation to work out a variable-coefficient-dependent auto-Bäcklund transformation, along with two variable-coefficient-dependent classes of the shock-wave-type solutions. Our auto-Bäcklund transformation is different from the recently reported bilinear one.

SYMBOLIC COMPUTATION FOR THE FISHER-TYPE EQUATION WITH VARIABLE COEFFICIENTS

2001 ◽  
Vol 12 (08) ◽  
pp. 1251-1259 ◽  
Author(s):  
YI-TIAN GAO ◽  
BO TIAN
Keyword(s):  
Symbolic Computation ◽  
Solitary Waves ◽  
Variable Coefficient ◽  
Bäcklund Transformation ◽  
Type Equation ◽  
Variable Coefficients ◽  
Backlund Transformation ◽  
Spatial Spread ◽  
Special Cases ◽  
Early Farming

For the variable-coefficient Fisher-type equation, which models such spatial spread as of an advantageous gene in a population or of early farming, we, in this paper, make use of computerized symbolic computation and report a new auto-Bäcklund transformation and a couple of new families of soliton-like solutions. Sample solitary waves are presented as the special cases.

A Truncated Painlevé Expansion and Exact Analytical Solutions for the Nonlinear Schr¨Odinger Equation with Variable Coefficients

2005 ◽  
Vol 60 (11-12) ◽  
pp. 768-774 ◽  
Author(s):  
Biao Li ◽  
Yong Chen
Keyword(s):  
Analytical Solutions ◽  
Bäcklund Transformation ◽  
Variable Coefficients ◽  
Backlund Transformation ◽  
Exact Analytical Solutions ◽  
Interaction Of Solitons ◽  
Varying Dispersion ◽  
Physical Phenomena ◽  
Truncated Painlevé Expansion ◽  
Singular Soliton

By using the truncated Painlevé expansion analysis an auto-Bäcklund transformation is found for the nonlinear Schrödinger equation with varying dispersion, nonlinearity, and gain or absorption. Then, based on the obtained auto-Bäcklund transformation and symbolic computation, we explore some explicit exact solutions including soliton-like solutions, singular soliton-like solutions, which may be useful to explain the corresponding physical phenomena. Further, the formation and interaction of solitons are simulated by computer. - PACS Nos.: 05.45.Yv, 02.30.Jr, 42.65.Tg

Multisoliton Excitations for the Kadomtsev-Petviashvili Equation

2006 ◽  
Vol 61 (1-2) ◽  
pp. 32-38 ◽  
Author(s):  
Zheng-Yi Ma ◽  
Guo-Sheng Hua ◽  
Chun-Long Zheng
Keyword(s):  
Bäcklund Transformation ◽  
Backlund Transformation ◽  
Electrons And Positrons ◽  
Straight Line ◽  
Petviashvili Equation ◽  
Truncated Painlevé Expansion ◽  
Multisoliton Solutions

By means of the standard truncated Painlevé expansion and a special Bäcklund transformation, some exact multisoliton solutions are derived for the Kadomtsev-Petviashvili equation. The evolution properties of the multisoliton excitations are investigated and some novel features or interesting behaviors are revealed. The results show that four straight-line solitons are annihilated or produced with the time increases, which is very similar to the completely nonelastic collision among electrons and positrons.

Soliton interactions and Bäcklund transformation for a (2+1)-dimensional variable-coefficient modified Kadomtsev-Petviashvili equation in fluid dynamics

2018 ◽  
Vol 32 (02) ◽  
pp. 1750170 ◽  
Author(s):  
Zi-Jian Xiao ◽  
Bo Tian ◽  
Yan Sun
Keyword(s):  
Fluid Dynamics ◽  
Shock Waves ◽  
Variable Coefficient ◽  
Bäcklund Transformation ◽  
Bilinear Forms ◽  
Backlund Transformation ◽  
Bell Polynomial ◽  
Soliton Interactions ◽  
Petviashvili Equation ◽  
Dimensional Variable

In this paper, we investigate a (2[Formula: see text]+[Formula: see text]1)-dimensional variable-coefficient modified Kadomtsev-Petviashvili (mKP) equation in fluid dynamics. With the binary Bell-polynomial and an auxiliary function, bilinear forms for the equation are constructed. Based on the bilinear forms, multi-soliton solutions and Bell-polynomial-type Bäcklund transformation for such an equation are obtained through the symbolic computation. Soliton interactions are presented. Based on the graphic analysis, Parametric conditions for the existence of the shock waves, elevation solitons and depression solitons are given, and it is shown that under the condition of keeping the wave vectors invariable, the change of [Formula: see text] and [Formula: see text] can lead to the change of the solitonic velocities, but the shape of each soliton remains unchanged, where [Formula: see text] and [Formula: see text] are the variable coefficients in the equation. Oblique elastic interactions can exist between the (i) two shock waves, (ii) two elevation solitons, and (iii) elevation and depression solitons. However, oblique interactions between (i) shock waves and elevation solitons, (ii) shock waves and depression solitons are inelastic.

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