On the stability of travelling wave solutions of a one-dimensional model system for compressible viscous gas

We investigate the stability of travelling wave solutions of the one-dimensional under-cooled Stefan problem. We find a necessary and sufficient condition on the initial datum under which the free boundary is asymptotic to a travelling wave front. The method applies also to other types of solutions.

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One- and two-dimensional travelling wave solutions in gas-fluidized beds

Journal of Fluid Mechanics ◽

10.1017/s0022112096001280 ◽

1996 ◽

Vol 306 ◽

pp. 183-221 ◽

Cited By ~ 53

Author(s):

B. J. Glasser ◽

I. G. Kevrekidis ◽

S. Sundaresan

Keyword(s):

Fluidized Beds ◽

Equations Of Motion ◽

High Amplitude ◽

Travelling Wave ◽

Numerical Continuation ◽

Model Parameters ◽

Travelling Wave Solutions ◽

Two Dimensional ◽

One Dimensional ◽

Wave Solutions

Making use of numerical continuation techniques as well as bifurcation theory, both one- and two-dimensional travelling wave solutions of the ensemble-averaged equations of motion for gas and particles in fluidized beds have been computed. One-dimensional travelling wave solutions having only vertical structure emerge through a Hopf bifurcation of the uniform state and two-dimensional travelling wave solutions are born out of these one-dimensional waves. Fully developed two-dimensional solutions of high amplitude are reminiscent of bubbles. It is found that the qualitative features of the bifurcation diagram are not affected by changes in model parameters or the closures. An examination of the stability of one-dimensional travelling wave solutions to two-dimensional perturbations suggests that two-dimensional solutions emerge through a mechanism which is similar to the overturning instability analysed by Batchelor & Nitsche (1991).

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

Proceedings of the Royal Society of Edinburgh Section A Mathematics ◽

10.1017/s0308210500024525 ◽

1990 ◽

Vol 115 (1-2) ◽

pp. 1-18 ◽

Cited By ~ 2

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.

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Existence of travelling-wave solutions representing domain wall motion in a thin ferromagnetic nanowire

Proceedings of the Royal Society of Edinburgh Section A Mathematics ◽

10.1017/s0308210517000130 ◽

2017 ◽

Vol 148 (2) ◽

pp. 395-407

Author(s):

Ross G. Lund ◽

J. M. Robbins ◽

Valeriy Slastikov

Keyword(s):

Domain Wall ◽

Wall Motion ◽

Domain Wall Motion ◽

Travelling Wave ◽

Travelling Wave Solutions ◽

Implicit Function ◽

One Dimensional ◽

Wave Solutions ◽

Static Solutions ◽

Ferromagnetic Nanowire

We study the dynamics of a domain wall under the influence of applied magnetic fields in a one-dimensional ferromagnetic nanowire, governed by the Landau–Lifshitz–Gilbert equation. Existence of travelling-wave solutions close to two known static solutions is proven using implicit-function-theorem-type arguments.

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