# Nonlinear Convective Instability of Turing-Unstable Fronts near Onset: A Case Study

@article{Ghazaryan2007NonlinearCI, title={Nonlinear Convective Instability of Turing-Unstable Fronts near Onset: A Case Study}, author={Anna Ghazaryan and Bj{\"o}rn Sandstede}, journal={SIAM J. Appl. Dyn. Syst.}, year={2007}, volume={6}, pages={319-347} }

Fronts are traveling waves in spatially extended systems that connect two different spatially homogeneous rest states. If the rest state behind the front undergoes a supercritical Turing instabilit...

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## References

SHOWING 1-10 OF 31 REFERENCES

Nonlinear Stability of Bifurcating Front Solutions for the Taylor-Couette Problem

- Mathematics
- 2000

We consider the Taylor-Couette problem in an infinitely extended cylindrical domain. There exist modulated front solutions which describe the spreading of the stable Taylor vortices into the region…

Nonlinear convective instability of fronts: a case study

- Mathematics
- 2005

We consider a model system, consisting of two nonlinearly coupled partial differential equations, to investigate nonlinear convective instabilities of travelling waves. The system exhibits front…

Hopf Bifurcation and Exchange of Stability in Diffusive Media

- Mathematics
- 2004

Abstract.We consider solutions bifurcating from a spatially homogeneous equilibrium under the assumption that the associated linearization possesses a continuous spectrum up to the imaginary axis,…

Stable transport of information near essentially unstable localized structures

- Mathematics
- 2003

When the steady states at infinity become unstable through a
pattern forming bifurcation, a travelling wave may bifurcate
into a modulated front which is time-periodic in a moving
frame. This…

Essential instabilities of fronts: bifurcation, and bifurcation failure

- Mathematics
- 2001

Various instability mechanisms of fronts in reaction-diffusion systems are analysed; the emphasis is on instabilities that have the potential to create modulated (i.e. time-periodic) waves near the…

Local stability of critical fronts in nonlinear parabolic partial differential equations

- Mathematics
- 1994

For the Ginzburg-Landau equation and similar nonlinear parabolic partial differential equations on the real line, we prove the nonlinear stability of the slowest monotonic front solution by computing…

Non-linear Stability of Modulated Fronts¶for the Swift–Hohenberg Equation

- Mathematics, Physics
- 2000

Abstract: We consider front solutions of the Swift–Hohenberg equation ∂tu= -(1+ ∂x2)2u + ɛ2u -u3. These are traveling waves which leave in their wake a periodic pattern in the laboratory frame. Using…

Spectral stability of modulated travelling waves bifurcating near essential instabilities

- Physics, MathematicsProceedings of the Royal Society of Edinburgh: Section A Mathematics
- 2000

Localized travelling waves to reaction-diffusion systems on the real line are investigated. The issue addressed in this work is the transition to instability which arises when the essential spectrum…