Painlevé Test, Generalized Symmetries, Bäcklund Transformations and Exact Solutions to the Third-Order Burgers’ Equations

2014 ◽  
Vol 158 (2) ◽  
pp. 433-446 ◽  
Author(s):  
Hanze Liu
Keyword(s):  
Exact Solutions ◽  
Bäcklund Transformations ◽  
Painlevé Test ◽  
Third Order ◽  
Burgers Equations ◽  
The Third ◽  
Generalized Symmetries ◽  
Backlund Transformations

A NEW PERSPECTIVE TO STUDY THE THIRD-ORDER MODIFIED KDV EQUATION ON FRACTAL SET

Fractals ◽  
2020 ◽  
Vol 28 (06) ◽  
pp. 2050110 ◽  
Author(s):  
JIAN-GEN LIU ◽  
XIAO-JUN YANG ◽  
YI-YING FENG ◽  
PING CUI
Keyword(s):  
Kdv Equation ◽  
Bäcklund Transformations ◽  
Nonlinear Phenomena ◽  
Third Order ◽  
Fractal Set ◽  
The Third ◽  
Backlund Transformations ◽  
Modified Kdv Equation ◽  
Constant Coefficients ◽  
New Perspective

In this paper, we construct the Bäcklund transformations and the super-position formulas to the constant coefficients local fractional Riccati equation for the first time. Next, by means of the Bäcklund transformations and seed solutions which have been known in [X. J. Yang et al., Non-differentiable solutions for local fractional nonlinear Riccati differential equations, Fundam. Inform. 151(1–4) (2017) 409–417], we can get a class of exact solutions to the third-order modified KdV equation on the fractal set. These new type solutions can assist us to review different nonlinear phenomena better, which had been modeled via local fractional derivative.

scholarly journals Bäcklund Transformation and New Exact Solutions of the Sharma-Tasso-Olver Equation

2011 ◽  
Vol 2011 ◽  
pp. 1-8 ◽  
Author(s):  
Lin Jianming ◽  
Ding Jie ◽  
Yuan Wenjun
Keyword(s):  
Exact Solutions ◽  
Bäcklund Transformation ◽  
Analytic Study ◽  
Bäcklund Transformations ◽  
Painlevé Analysis ◽  
Backlund Transformation ◽  
New Exact Solutions ◽  
Backlund Transformations ◽  
Painleve Analysis

The Sharma-Tasso-Olver (STO) equation is investigated. The Painlevé analysis is efficiently used for analytic study of this equation. The Bäcklund transformations and some new exact solutions are formally derived.

Painlevé Test, Bäcklund Transformation and Consistent Riccati Expansion Solvability for two Generalised Cylindrical Korteweg-de Vries Equations with Variable Coefficients

2018 ◽  
Vol 73 (3) ◽  
pp. 207-213 ◽  
Author(s):  
Rehab M. El-Shiekh
Keyword(s):  
Vries Equation ◽  
Dusty Plasmas ◽  
Variable Coefficients ◽  
Bäcklund Transformations ◽  
Painlevé Test ◽  
New Exact Solutions ◽  
Backlund Transformations ◽  
Integrability Conditions ◽  
De Vries ◽  
Consistent Riccati Expansion

AbstractIn this paper, the integrability of the (2+1)-dimensional cylindrical modified Korteweg-de Vries equation and the (3+1)-dimensional cylindrical Korteweg-de Vries equation with variable coefficients arising in dusty plasmas in its generalised form was studied by two different techniques: the Painlevé test and the consistent Riccati expansion solvability. The integrability conditions and Bäcklund transformations are constructed. By using Bäcklund transformations and the solutions of the Riccati equation many new exact solutions are found for the two equations in this study. Finally, the application of the obtained solutions in dusty plasmas is investigated.

scholarly journals Third-Order Conditional Lie–Bäcklund Symmetries of Nonlinear Reaction-Diffusion Equations

2017 ◽  
Vol 2017 ◽  
pp. 1-9 ◽  
Author(s):  
Keqin Su ◽  
Jie Cao
Keyword(s):  
Exact Solutions ◽  
Reaction Diffusion ◽  
Reaction Diffusion Equations ◽  
Diffusion Equations ◽  
Governing Equations ◽  
Third Order ◽  
Differential Constraints ◽  
The Third ◽  
Nonlinear Reaction

The third-order conditional Lie–Bäcklund symmetries of nonlinear reaction-diffusion equations are constructed due to the method of linear determining equations. As a consequence, the exact solutions of the resulting equations are derived due to the compatibility of the governing equations and the admitted differential constraints, which are resting on the characteristic of the admitted conditional Lie–Bäcklund symmetries to be zero.

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