Solitary and Travelling Wave Solutions of Certain Nonlinear Diffusion-Reaction Equations

2020 ◽  
Author(s):  
Miftachul Hadi
Keyword(s):  
Nonlinear Diffusion ◽  
Travelling Wave ◽  
Diffusion Reaction ◽  
Travelling Wave Solutions ◽  
Wave Solutions ◽  
Reaction Equations

We review the work of Ranjit Kumar, R S Kaushal, Awadhesh Prasad. The work is still in progress.

scholarly journals Travelling wave solutions in a negative nonlinear diffusion–reaction model

2020 ◽  
Vol 81 (6-7) ◽  
pp. 1495-1522
Author(s):  
Yifei Li ◽  
Peter van Heijster ◽  
Robert Marangell ◽  
Matthew J. Simpson
Keyword(s):  
Nonlinear Diffusion ◽  
Wave Speed ◽  
Geometric Approach ◽  
Travelling Wave ◽  
Reaction Model ◽  
Diffusion Reaction ◽  
Travelling Wave Solutions ◽  
Wave Solutions ◽  
Minimum Wave Speed ◽  
Nonlinear Diffusion Reaction Equation

AbstractWe use a geometric approach to prove the existence of smooth travelling wave solutions of a nonlinear diffusion–reaction equation with logistic kinetics and a convex nonlinear diffusivity function which changes sign twice in our domain of interest. We determine the minimum wave speed, $$c^*$$ c ∗ , and investigate its relation to the spectral stability of a desingularised linear operator associated with the travelling wave solutions.

scholarly journals Diffusive waves in inhomogeneous media

1989 ◽  
Vol 32 (2) ◽  
pp. 291-315 ◽  
Author(s):  
Paul C. Fife
Keyword(s):  
Diffusion Equation ◽  
Nonlinear Diffusion ◽  
Monotone Function ◽  
Travelling Wave ◽  
Inhomogeneous Media ◽  
Nonlinear Diffusion Equation ◽  
Travelling Wave Solutions ◽  
Wave Solutions ◽  
Stable Solutions ◽  
Image Position

When the function f(u) is of “bistable type’, i.e. has two zeros h̲ and h+ at which f' is negative and (for simplicity) has only one other zero between them, then the constant functions u = h± are L∞-stable solutions of the nonlinear diffusion equationIn addition, there are travelling wave solutions u+(x, t) and u̲(x, t) which, ifconnect h+ to h̲ in the sense thatthe convergence being uniform on bounded x-intervals. These solutions are of the formwhere U(z) is a monotone function (the wave's profile), U(±∞) = h±, and the velocity c is a specific positive number depending on the function f.

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