Abstract
The effect of distant endwalls on a Hopf bifurcation from a translation-invariant state is considered. The walls break the translation symmetry with the result that the initial bifurcation is to standing waves with a fixed phase. Travelling waves (TW) appear in a secondary pitchfork bifurcation. A new two-frequency state (MW) is present at small amplitude only. The theory is applied to systems, such as binary fluid convection, that are described by coupled complex Ginzburg-Landau equations. As a result the TW and MW are tentatively identified with the confined travelling wave states and the blinking states, respectively, observed both experimentally and numerically in systems with sufficiently large group velocity.
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
R.W. Waiden, P. Kolodner, A. Passner and C.M. Surko, Traveling waves and chaos in convection in binary fluid mixtures, Phys. Rev. Lett. 55: 496 (1985).
E. Moses, J. Fineberg and V. Steinberg, Multistability and confined traveling-wave patterns in a convecting binary mixture, Phys. Rev. A 35: 2757 (1987);
R. Heinrichs, G. Ahlers and D.S. Cannell, Traveling waves and spatial variation in the convection of a binary mixture, Phys. Rev. A 35: 2761 (1987).
J. Fineberg, E. Moses and V. Steinberg, Spatially and temporally modulated traveling-wave pattern in convecting binary mixtures, Phys. Rev. Lett. 61: 838 (1988);
P. Kolodner and C.M. Surko, Weakly nonlinear traveling-wave convection, Phys. Rev. Lett. 61: 842 (1988).
A.E. Deane, E. Knobloch, J. Toomre, Traveling waves and chaos in thermosolutal convection, Phys. Rev. A 36: 2862 (1987);
S.J. Linz, M. Lücke, H.W. Müller and J. Niederländer, Convection in binary fluid mixtures: Traveling waves and lateral currents, Phys. Rev. A 38: 5727 (1988).
C Bretherton and E.A. Spiegel, Intermittency through modulational instability, Phys. Lett. 96 A: 152 (1983); E. Knobloch, Doubly diffusive waves, in Doubly Diffusive Motions, FED-Vol. 24, N.E. Bixler and E.A. Spiegel, eds., ASME, New York (1985).
M.C Cross, Traveling and standing waves in binary-fluid convection in finite geometries, Phys. Rev. Lett. 57: 2935 (1986).
A.E. Deane, E. Knobloch and J. Toomre, Doubly diffusive waves, in Proc. Int. Conf. on Fluid Mechanics (Beijing 1987), Peking University Press (1987);
A.E. Deane, E. Knobloch and J. Toomre, Traveling waves in large-aspect-ratio thermosolutal convection, Phys. Rev. A 37: 1817 (1988).
M.C. Cross, Structure of nonlinear traveling-wave states in finite geometries, Phys. Rev. A 38: 3593 (1988).
P. Coullet, S. Fauve and E. Tirapegui, Large scale instability of nonlinear standing waves, J. Phys. Lettres 46: L-787 (1985).
M. Golubitsky and I. Stewart, Hopf bifurcation in the presence of symmetry, Arch. Rat. Mech. and Anal. 87: 107 (1985).
E. Knobloch, Oscillatory convection in binary mixtures, Phys. Rev. A 34: 1538 (1986).
M.C Cross and K. Kim, Linear instability and the codimension-2 region in binary fluid convection between rigid impermeable boundaries, Phys. Rev. A 37: 3909 (1988).
W. Schöpf and W. Zimmermann, Multicritical behaviour in binary fluid convection, Eu-rophys. Lett. 8: 41 (1989).
G. Dangelmayr and D. Armbruster, Steady state mode interactions in the presence of 0(2) symmetry and in non-flux boundary conditions, Contemp. Math. 56: 53 (1986).
J.D. Crawford, M. Golubitsky, M.G.M. Gomas, E. Knobloch and I. Stewart, Boundary conditions as symmetry constraints, in Proc. Warwick Symp. on Singularity Theory and its Applications, to appear (1990).
M.C. Cross, P.G. Daniels, P.C. Hohenberg and E.D. Siggia, Phase-winding solutions in a finite container above the convective threshold, J. Fluid Mech. 127: 155 (1983); see also ref. [8].
G. Dangelmayr and E. Knobloch, On the Hopf bifurcation with broken O(2) symmetry, in The Physics of Structure Formation: Theory and Simulation, W. Güttinger and G. Dangelmayr, eds., Springer-Verlag, Berlin (1987);
G. Dangelmayr and E. Knobloch, Hopf bifurcation in reaction-diffusion equations with broken translation symmetry, in Proc. of the Int. Conf. on Bifurcation Theory and its Numerical Anlaysis, Li Kaitai, J. Marsden, M. Golubitsky, G. Iooss, eds., Xian Jiatong University Press, Xian (1989).
G. Dangelmayr and E. Knobloch, Hopf bifurcation with broken circular symmetry, Non-linearity (submitted).
S. van Gils and J. Mallet-Paret, Hopf bifurcation and symmetry: travelling and standing waves on the circle, Proc. Roy. Soc. Edinburgh 104 A: 279 (1986).
E. Knobloch, On the degenerate Hopf bifurcation with 0(2) symmetry, Contemp. Math. 56: 193 (1986).
J.D. Crawford and E. Knobloch, On degenerate Hopf bifurcation with broken O(2) symmetry, Nonlinearity 1: 617 (1988).
T. Ohta and K. Kawasaki, Euclidean invariant phase dynamics for propagating pattern, Physica 27 D: 21 (1987);
M. Bestehorn, R. Friedrich and H. Haken, The oscillatory instability of a spatially homogeneous state in large aspect ratio systems of fluid dynamics, Z. Phys. B 72: 265 (1988).
M. Bestehorn, R. Friedrich and H. Haken, Two-dimensional traveling wave patterns in nonequilibrium systems, Z. Phys. B 75: 265 (1989).
Author information
Authors and Affiliations
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 1990 Plenum Press, New York
About this chapter
Cite this chapter
Dangelmayr, G., Knobloch, E. (1990). Dynamics of Slowly Varying Wavetrains in Finite Geometry. In: Busse, F.H., Kramer, L. (eds) Nonlinear Evolution of Spatio-Temporal Structures in Dissipative Continuous Systems. NATO ASI Series, vol 225. Springer, Boston, MA. https://doi.org/10.1007/978-1-4684-5793-3_39
Download citation
DOI: https://doi.org/10.1007/978-1-4684-5793-3_39
Publisher Name: Springer, Boston, MA
Print ISBN: 978-1-4684-5795-7
Online ISBN: 978-1-4684-5793-3
eBook Packages: Springer Book Archive