scholarly journals Asymptotic nonlinear stability of traveling waves to a system of coupled Burgers equations

2013 ◽  
Vol 397 (1) ◽  
pp. 322-333 ◽  
Author(s):  
Yanbo Hu
Keyword(s):  
Nonlinear Stability ◽  
Traveling Waves ◽  
Burgers Equations

scholarly journals Hyperbolic traveling waves driven by growth

2014 ◽  
Vol 24 (06) ◽  
pp. 1165-1195 ◽  
Author(s):  
Emeric Bouin ◽  
Vincent Calvez ◽  
Grégoire Nadin
Keyword(s):  
Nonlinear Stability ◽  
Traveling Waves ◽  
Hyperbolic Model ◽  
Kpp Equation ◽  
Transition Parameter ◽  
Minimal Speed ◽  
Traveling Fronts ◽  
Traveling Front ◽  
Maximal Speed ◽  
Existence And Stability

We perform the analysis of a hyperbolic model which is the analog of the Fisher-KPP equation. This model accounts for particles that move at maximal speed ϵ-1 (ϵ > 0), and proliferate according to a reaction term of monostable type. We study the existence and stability of traveling fronts. We exhibit a transition depending on the parameter ϵ: for small ϵ the behavior is essentially the same as for the diffusive Fisher-KPP equation. However, for large ϵ the traveling front with minimal speed is discontinuous and travels at the maximal speed ϵ-1. The traveling fronts with minimal speed are linearly stable in weighted L2 spaces. We also prove local nonlinear stability of the traveling front with minimal speed when ϵ is smaller than the transition parameter.

EVANS FUNCTIONS AND NONLINEAR STABILITY OF TRAVELING WAVES IN NEURONAL NETWORK MODELS

2007 ◽  
Vol 17 (08) ◽  
pp. 2693-2704 ◽  
Author(s):  
BJÖRN SANDSTEDE
Keyword(s):  
Differential Equations ◽  
Neuronal Network ◽  
Nonlinear Stability ◽  
Traveling Waves ◽  
Network Models ◽  
Function Spaces ◽  
Spectral Stability ◽  
Evans Function ◽  
Evans Functions

Modeling networks of synaptically coupled neurons often leads to systems of integro-differential equations. Particularly interesting solutions in this context are traveling waves. We prove here that spectral stability of traveling waves implies their nonlinear stability in appropriate function spaces, and compare several recent Evans-function constructions that are useful tools when analyzing spectral stability.

NONLINEAR STABILITY OF PLANAR STATIONARY WAVES FOR GENERALIZED BENJAMIN–BONA–MAHONY–BURGERS EQUATIONS IN THE HALF-PLANE

2013 ◽  
Vol 10 (02) ◽  
pp. 283-333 ◽  
Author(s):  
QINGHUA XIAO ◽  
HUIJIANG ZHAO
Keyword(s):  
Half Plane ◽  
Nonlinear Stability ◽  
Convergence Rates ◽  
Type Inequality ◽  
Global Solutions ◽  
Stationary Waves ◽  
Hardy Type Inequality ◽  
Weighted Energy Method ◽  
Burgers Equations ◽  
The Best Possible Constant

This paper is concerned with the nonlinear stability of planar stationary waves of generalized Benjamin–Bona–Mahony–Burgers equations in half-plane. The planar stationary waves are shown to be globally nonlinear stable and then, by employing the space–time weighted energy method developed by Kawashima and Matsumura in 1995 and a generalized Hardy type inequality with the best possible constant introduced by Kawashima and Kurata in 2009, the corresponding convergence rates (both algebraic and exponential) of the global solutions to the stationary waves are also obtained for both the nondegenerate case and the degenerate case.

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