Nonlinear Evolution of Spatio-Temporal Structures in Dissipative Continuous Systems pp 399-410 | Cite as

# Dynamics of Slowly Varying Wavetrains in Finite Geometry

## 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.

## Keywords

Rayleigh Number Hopf Bifurcation Standing Wave Bifurcation Diagram Travel Wave Solution## Preview

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## References

- [1]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).ADSCrossRefGoogle Scholar - [2]E. Moses, J. Fineberg and V. Steinberg,
*Multistability and confined traveling-wave patterns in a convecting binary mixture*, Phys. Rev. A**35**: 2757 (1987);ADSCrossRefGoogle Scholar - [2a]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).ADSCrossRefGoogle Scholar - [3]J. Fineberg, E. Moses and V. Steinberg,
*Spatially and temporally modulated traveling-wave pattern in convecting binary mixtures*, Phys. Rev. Lett.**61**: 838 (1988);ADSCrossRefGoogle Scholar - [3a]P. Kolodner and C.M. Surko,
*Weakly nonlinear traveling-wave convection*, Phys. Rev. Lett.**61**: 842 (1988).ADSCrossRefGoogle Scholar - [4]A.E. Deane, E. Knobloch, J. Toomre,
*Traveling waves and chaos in thermosolutal convection*, Phys. Rev. A**36**: 2862 (1987);ADSCrossRefGoogle Scholar - [4a]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).ADSGoogle Scholar - [5]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).Google Scholar - [6]M.C Cross,
*Traveling and standing waves in binary-fluid convection in finite geometries*, Phys. Rev. Lett.**57**: 2935 (1986).ADSCrossRefGoogle Scholar - [7]A.E. Deane, E. Knobloch and J. Toomre,
*Doubly diffusive waves*, in*Proc. Int. Conf. on Fluid Mechanics (Beijing 1987)*, Peking University Press (1987);Google Scholar - [7a]A.E. Deane, E. Knobloch and J. Toomre,
*Traveling waves in large-aspect-ratio thermosolutal convection*, Phys. Rev.**A 37**: 1817 (1988).ADSGoogle Scholar - [8]M.C. Cross,
*Structure of nonlinear traveling-wave states in finite geometries*, Phys. Rev.**A 38**: 3593 (1988).MathSciNetADSGoogle Scholar - [9]P. Coullet, S. Fauve and E. Tirapegui,
*Large scale instability of nonlinear standing waves*, J. Phys. Lettres**46**: L-787 (1985).Google Scholar - [10]M. Golubitsky and I. Stewart,
*Hopf bifurcation in the presence of symmetry*, Arch. Rat. Mech. and Anal.**87**: 107 (1985).MathSciNetADSMATHCrossRefGoogle Scholar - [11]E. Knobloch,
*Oscillatory convection in binary mixtures*, Phys. Rev.**A 34**: 1538 (1986).ADSGoogle Scholar - [12]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).MathSciNetADSGoogle Scholar - [13]W. Schöpf and W. Zimmermann,
*Multicritical behaviour in binary fluid convection*, Eu-rophys. Lett.**8**: 41 (1989).ADSCrossRefGoogle Scholar - [14]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).MathSciNetCrossRefGoogle Scholar - [15]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).Google Scholar - [16]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].MathSciNetADSMATHCrossRefGoogle Scholar - [17]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);Google Scholar - [17a]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).Google Scholar - [18]G. Dangelmayr and E. Knobloch,
*Hopf bifurcation with broken circular symmetry*, Non-linearity (submitted).Google Scholar - [19]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).CrossRefGoogle Scholar - [20]E. Knobloch,
*On the degenerate Hopf bifurcation with 0(2) symmetry*, Contemp. Math.**56**: 193 (1986).MathSciNetCrossRefGoogle Scholar - [21]J.D. Crawford and E. Knobloch,
*On degenerate Hopf bifurcation with broken O(2) symmetry*, Nonlinearity**1**: 617 (1988).MathSciNetADSMATHCrossRefGoogle Scholar - [22]T. Ohta and K. Kawasaki,
*Euclidean invariant phase dynamics for propagating pattern*, Physica**27 D**: 21 (1987);MathSciNetADSGoogle Scholar - [22a]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).ADSCrossRefGoogle Scholar - [23]M. Bestehorn, R. Friedrich and H. Haken,
*Two-dimensional traveling wave patterns in nonequilibrium systems*, Z. Phys.**B 75**: 265 (1989).ADSCrossRefGoogle Scholar