Abstract
The behaviour of pattern-forming systems in one dimension with Galilean symmetry in large domains is not described by the usual Ginzburg-Landau equation. This is because the Galilean symmetry leads to a large-scale neutral mode that interacts with the pattern. The resulting coupled amplitude equations, derived by considering the symmetry, show chaotic behaviour and exhibit a novel scaling in which the amplitude of the pattern is proportional to the 3/4 power of the bifurcation parameter.
This is a preview of subscription content, log in via an institution to check access.
Access this chapter
Tax calculation will be finalised at checkout
Purchases are for personal use only
Preview
Unable to display preview. Download preview PDF.
References
P. Barthelet and F. Charru (1998) Benjamin-Feir and Eckhaus instabilities with Galilean invariance: the case of interfacial waves in viscous shear flows. Eur. J. Mech. B 17, 1–18.
I.A. Beresnov and V.N. Nikolaevskiy (1993) A model for nonlinear seismic waves in a medium with instability. Physica D 66, 1–6.
P. Coullet and S. Fauve (1985) Propagative phase dynamics for systems with Galilean invariance. Phys. Rev. Lett. 55, 2857–2859.
S.M. Cox and P.C. Matthews (2001) Exponential time differencing for stiff systems. J. Comp. Physics, submitted.
M. Golubitsky, I. Stewart and D.A. Schaeffer (1988) Singularities and groups in bifurcation theory. Volume II. Appl. Math. Sci. 69, Springer-Verlag, New York.
I.L. Kliakhandler and B.A. Malomed (1997) Short-wavelength instability in presence of a zero mode: anomalous growth law. Phys. Lett. A 231, 191–194.
P.C. Matthews and S.M. Cox (2000) One-dimensional pattern formation with Galilean invariance near a stationary bifurcation. Phys. Rev. E 62, R1473–R1476.
P.C. Matthews and S.M. Cox (2000) Pattern formation with a conservation law. Nonlinearity 13, 1293–1320.
I. Melbourne (1999) Steady state bifurcation with Euclidean symmetry. Trans. Amer. Math. Soc. 351, 1575–1603.
M.I. Tribelsky and K. Tsuboi (1996) New scenario for transition to turbulence? Phys. Rev. Lett. 76, 1631–1634.
Author information
Authors and Affiliations
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 2003 Springer Basel AG
About this chapter
Cite this chapter
Matthews, P.C., Cox, S.M. (2003). Pattern Formation with Galilean Symmetry. In: Buescu, J., Castro, S.B.S.D., da Silva Dias, A.P., Labouriau, I.S. (eds) Bifurcation, Symmetry and Patterns. Trends in Mathematics. Birkhäuser, Basel. https://doi.org/10.1007/978-3-0348-7982-8_15
Download citation
DOI: https://doi.org/10.1007/978-3-0348-7982-8_15
Publisher Name: Birkhäuser, Basel
Print ISBN: 978-3-0348-9642-9
Online ISBN: 978-3-0348-7982-8
eBook Packages: Springer Book Archive