A note on the stability of travelling wave solutions of Burgers’ equation

1985 ◽  
Vol 2 (1) ◽  
pp. 27-35 ◽  
Author(s):  
Kenji Nishihara
Keyword(s):  
Burgers Equation ◽  
Travelling Wave ◽  
Travelling Wave Solutions ◽  
Wave Solutions ◽  
The Stability

Travelling-wave solutions to the Korteweg-de Vries-Burgers equation

1985 ◽  
Vol 101 (3-4) ◽  
pp. 207-226 ◽  
Author(s):  
J. L. Bona ◽  
M. E. Schonbek
Keyword(s):  
Burgers Equation ◽  
Travelling Wave ◽  
Singular Limit ◽  
Travelling Wave Solutions ◽  
Qualitative Properties ◽  
Wave Solutions ◽  
De Vries ◽  
Limiting Behaviour ◽  
Image Position

SynopsisThe existence and certain qualitative properties of travelling-wave solutions to the Korteweg-de Vries-Burgers equation,are established. The limiting behaviour of these waves, when ε tends to zero and when δ tends to zero is examined together with a singular limit wherein both ε and δ tend to zero.

Existence and stability of travelling wave solutions for an evolutionary ecology model

1990 ◽  
Vol 115 (1-2) ◽  
pp. 1-18 ◽  
Author(s):  
Roger Lui
Keyword(s):  
Population Size ◽  
Evolutionary Ecology ◽  
Reaction Diffusion ◽  
Travelling Wave ◽  
Travelling Wave Solutions ◽  
Fisher’S Equation ◽  
Wave Solutions ◽  
Fisher's Equation ◽  
Existence And Stability ◽  
The Stability

SynopsisMonotone travelling wave solutions are known to exist for Fisher's equation which models the propagation of an advantageous gene in a single locus, two alleles population genetics model. Fisher's equation assumed that the population size is a constant and that the fitnesses of the individuals in the population depend only on their genotypes. In this paper, we relax these assumptions and allow the fitnesses to depend also on the population size. Under certain assumptions, we prove that in the second heterozygote intermediate case, there exists a constant θ*>0 such that monotone travelling wave solutions for the reaction–diffusion system exist whenever θ > θ*. We also discuss the stability properties of these waves.

Travelling Wave Solutions for the Burgers Equation and the Korteweg-de Vries Equation with Variable Coefficients Using the Generalized (G´/G)-Expansion Method

2010 ◽  
Vol 65 (12) ◽  
pp. 1065-1070 ◽  
Author(s):  
Elsayed M. E. Zayed ◽  
Mahmoud A. M. Abdelaziz
Keyword(s):  
Burgers Equation ◽  
Kdv Equation ◽  
Expansion Method ◽  
Travelling Wave ◽  
Variable Coefficients ◽  
Nonlinear Evolution Equations ◽  
Hyperbolic Function ◽  
Travelling Wave Solutions ◽  
Wave Solutions ◽  
De Vries

In this article, a generalized (G´/G)-expansion method is used to find exact travelling wave solutions of the Burgers equation and the Korteweg-de Vries (KdV) equation with variable coefficients. As a result, hyperbolic, trigonometric, and rational function solutions with parameters are obtained. When these parameters are taking special values, the solitary wave solutions are derived from the hyperbolic function solution. It is shown that the proposed method is direct, effective, and can be applied to many other nonlinear evolution equations in mathematical physics.

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