# Modulating traveling fronts in a dispersive Swift-Hohenberg equation coupled to an additional conservation law

@inproceedings{Hilder2021ModulatingTF, title={Modulating traveling fronts in a dispersive Swift-Hohenberg equation coupled to an additional conservation law}, author={Bastian Hilder}, year={2021} }

We consider a one-dimensional Swift-Hohenberg equation coupled to a conservation law, where both equations contain additional dispersive terms breaking the reflection symmetry x 7→ −x. This system exhibits a Turing instability and we study the dynamics close to the onset of this instability. First, we show that periodic traveling waves bifurcate from a homogeneous ground state. Second, fixing the bifurcation parameter close to the onset of instability, we construct modulating traveling fronts…

## References

SHOWING 1-10 OF 27 REFERENCES

Modulating traveling fronts for the Swift-Hohenberg equation in the case of an additional conservation law

- Mathematics
- 2018

We consider the one-dimensional Swift-Hohenberg equation coupled to a conservation law. As a parameter increases the system undergoes a Turing bifurcation. We study the dynamics near this…

Modulated traveling fronts for a nonlocal Fisher-KPP equation: a dynamical systems approach

- Mathematics
- 2014

We consider a nonlocal generalization of the Fisher-KPP equation in one spatial dimension. As a parameter is varied the system undergoes a Turing bifurcation. We study the dynamics near this Turing…

Spectral stability of pattern-forming fronts in the complex Ginzburg–Landau equation with a quenching mechanism

- Physics, MathematicsNonlinearity
- 2021

We consider pattern-forming fronts in the complex Ginzburg–Landau equation with a traveling spatial heterogeneity which destabilises, or quenches, the trivial ground state while progressing through…

Justification of the Ginzburg–Landau Approximation in Case of Marginally Stable Long Waves

- Mathematics, Computer ScienceJ. Nonlinear Sci.
- 2011

A method to handle a situation when the Ginzburg–Landau equation is violated by an additional curve of stable eigenvalues which possesses a vanishing real part at the Fourier wave number k=0 for all values of the bifurcation parameter is developed.

Bifurcating fronts for the Taylor-Couette problem in infinite cylinders

- Mathematics
- 1999

We show the existence of bifurcating fronts for the weakly unstable Taylor—Couette problem in an infinite cylinder. These fronts connect a stationary bifurcating pattern, here the Taylor vortices,…

Pattern forming pulled fronts: bounds and universal convergence

- Mathematics, Physics
- 2003

We analyze the dynamics of pattern forming fronts which propagate into an unstable state, and whose dynamics is of the pulled type, so that their asymptotic speed is equal to the linear spreading…

Swift-Hohenberg equation with broken reflection symmetry.

- Mathematics, MedicinePhysical review. E, Statistical, nonlinear, and soft matter physics
- 2009

It is found that the localized states now drift, and it is shown that the snakes-and-ladders structure breaks up into a stack of isolas, and the dynamics of the system outside of the snaking region is studied.

Pattern formation with a conservation law

- Mathematics, Physics
- 2000

Pattern formation in systems with a conserved quantity is considered by studying the appropriate amplitude equations. The conservation law leads to a
large-scale neutral mode that must be included…

Diffusive mixing of periodic wave trains in reaction–diffusion systems

- Physics, Mathematics
- 2011

Abstract We consider reaction–diffusion systems on the infinite line that exhibit a family of spectrally stable spatially periodic wave trains u 0 ( k x − ω t ; k ) that are parameterized by the wave…

The Dynamics of Modulated Wave Trains

- Physics, Mathematics
- 2005

The authors of this title investigate the dynamics of weakly-modulated nonlinear wave trains. For reaction-diffusion systems and for the complex Ginzburg - Landau equation, they establish rigorously…