Methods of Nonlinear Analysis D-12
Winter 2000
Hermann Riecke
Suggestions for Projects
- The impact of hidden symmetries on bifurcations [1,2,3]
- O(2)-Hopf bifurcation with weakly broken translation symmetry
[4,5,6,7]
- Dynamics of dislocations in steady patterns
within two-dimensional Ginzburg-Landau equation [8,9,10].
- The origin and justification of order-parameter
models of the Swift-Hohenberg type. [11,12]
- Pinning of structures and non-adiabatic effects, the main short-coming of the
separation of scales and the Ginzburg-Landau equation.
[13,14,15,16,17,18]
- Localized solutions of the CGL and their role in spatio-temporally chaotic solutions
[19,20,21,22,23,24,25]
- Bounded localized solutions of the 1-d CGL without saturation
[26,27,28,29]
- Coupling of bifurcating modes to weakly damped modes: mean flow in
Rayleigh-Bénard convection, concentration field in binary-mixture convection.
[30,31,32]
- Front propagation into unstable states: velocity selection and the marginal
stability criterion. [33,34,35]
- Instabilities of propagating fronts connecting two stable states, transition
from stationary to traveling fronts,
spot replication. [36,37,38]
- Interaction of localized structures, irregular sequences of pulses
[39,40,41]
- Long-wave equations: convection with poorly conducting boundaries
[42]
References
- [1]
-
J.D. Crawford.
Surface waves in non-square containers with square symmetry.
Phys. Rev. Lett., 67:441, 1991.
- [2]
-
J.D. Crawford, J.P. Gollub, and D. Lane.
Hidden symmetries of parametrically forced waves.
Nonlinearity, 6:119, 1993.
- [3]
-
D. Krmpotic, G.B. Mindlin, and C. Perez-Garcia.
Bénard-marnagoni convection in square containers.
preprint, 1996.
- [4]
-
G. Dangelmayr and E. Knobloch.
On the hopf bifurcation with broken o(2) symmetry.
In W. Güttinger and G. Dangelmayr, editors, The Physics of
Structure Formation, New York, 1987. Springer.
- [5]
-
G. Dangelmayr, E. Knobloch, and M. Wegelin.
Travelling wave convection in finite containers.
Proc. IUTAM on "Nonlinear Hydrodynamic Stability and
Transition", Europ. J. Mech. B, 1990.
- [6]
-
G. Dangelmayr and E. Knobloch.
On the hopf bifurcation with broken circular symmetry.
Nonlinearity, 4 (1991) 399, 4:399, 1991.
- [7]
-
P. Hirschberg and E. Knobloch.
Complex dynamics in the hopf bifurcation with broken translation
symmetry.
Physica D, 90:56, 1996.
- [8]
-
J.C. Neu.
Vortices in complex scalar fields.
Physica D, 43:385, 1990.
- [9]
-
E. Bodenschatz, W. Pesch, and L. Kramer.
Structure and dynamics of dislocations in anisotropic pattern-forming
systems.
Physica D, 32:135-145, 1988.
- [10]
-
E. Bodenschatz, A. Weber, and L. Kramer.
Interaction and dynamics of defects in convective roll patterns of
anisotropic fluids.
J. Stat. Phys., 64:1007, 1991.
- [11]
-
J. Swift and P.C. Hohenberg.
Hydrodynamic fluctuations at the convective instability.
Phys. Rev. A, 15:319, 1977.
- [12]
-
M. Bestehorn and R. Friedrich.
Rotationally invariant order parameter equations for natural patterns
in nonequilibrium systems.
Phys. Rev. E, 59:2642, 1999.
- [13]
-
M.C. Cross.
Wave-number selection by soft boundaries near threshold.
Phys. Rev. A, 29:391, 1984.
- [14]
-
Y. Pomeau.
Front motion, metastability and subcritical bifurcations in
hydrodynamics.
Physica D, 23:3, 1986.
- [15]
-
D. Bensimon, B.I. Shraiman, and V. Croquette.
Nonadiabatic effects in convection.
Phys. Rev. A, 38:5461, 1988.
- [16]
-
Y. Tu.
Worm structure in modified Swift-Hohenberg equation for
electroconvection.
Phys. Rev. E, 56:3765, 1997.
- [17]
-
C. Crawford and H. Riecke.
Oscillon-type structures and their interaction in a
Swift-Hohenberg equation.
Physica D, 129:83-92, 1999.
- [18]
-
David S. Cannell, M. A. Dominguez-Lerma, and Guenter Ahlers.
Experiments on wave number selection in rotating Couette-Taylor
flow.
Phys. Rev. Lett., 50:1365, 1983.
- [19]
-
K. Nozaki and N. Bekki.
J. Phys. Soc. Jpn. 53 (1984) 1581, 53:1581, 1984.
- [20]
-
Nozaki and Bekki.
Phys. Lett., 110:113, 1985.
- [21]
-
B. Janiaud, S. Jucquois, J. Lega, and V.Croquette.
Experimental evidence of bekki nozaki holes.
in Kai book (proceedings).
- [22]
-
H. Chate.
Stability of the bekki-nozaki hole solutions to the one-dimensional
complex ginzburg-landau equation.
Phys. Lett. A, 171:183, 1992.
- [23]
-
M. van Hecke.
The building blocks of spatiotemporal intermittency.
chao-dyn/9707010.
- [24]
-
J. Burguete, H. Chate, F. Daviaud, and N. Mukolobwiez.
Bekki-nozaki amplitude holes in hydrothermal nonlinear waves.
Phys. Rev. Lett., 82(16):3252-3255, April 1999.
- [25]
-
Lutz Brusch, Martin Zimmermann, Martin van Hecke, Markus Baer, and Alessandro
Torcini.
Modulated amplitude waves and the transition from phase to defect
chaos.
nlin.CD/0001068, 2000.
- [26]
-
C.S.Bretherton and E.A.Spiegel.
Intermittency through modulational instability.
Phys. Lett., 96:152, 1983.
- [27]
-
W. Schöpf and L. Kramer.
Small-amplitude periodic and chaotic solutions of the compex
ginzburg-landau equation for a subcritical bifurcation.
Phys. Rev. Lett., 66:2316, 1991.
- [28]
-
P. Kolodner, J.A. Glazier, and H. Williams.
Dispersive chaos in one-dimensional traveling-wave convection.
Phys. Rev. Lett., 65:1579, 1990.
- [29]
-
S. Popp, O. Stiller, E. Kuznetsov, and L. Kramer.
The cubic complex Ginzburg-Landau equation for a backward
bifurcation.
Physica D, 114:81, 1998.
- [30]
-
E.D. Siggia and A. Zippelius.
Pattern selection in Rayleigh-Béard convection near threshold.
Phys. Rev. Lett., 47:835, 1981.
- [31]
-
A. Bernoff.
Finite amplitude convection between stress-free boundaries;
ginzburg-landau equations and modulation theory.
Euro. J. Appl. Math., 5:267, 1994.
- [32]
-
H. Riecke.
Ginzburg-Landau equation coupled to a concentration field in
binary-mixture convection.
Physica D, 61:253-259, 1992.
- [33]
-
W. van Saarloos.
Front propagation into unstable states. ii. linear vs. nonlinear
marginal stability and rate of convergence.
Phys. Rev. A, 39:6367, 1989.
- [34]
-
W. van Saarloos, M. van Hecke, and R. Holyst.
Front propagation into unstable and metastable states in smectic-c*
liquid crystals: linear and nonlinear marginal-stability analysis.
Phys. Rev. E, 52:1773, 1995.
- [35]
-
W. van Saarloos.
Three basic issues concerning interface dynamics in nonequilibrium
pattern formation.
Phys. Rep., 301:9, 1998.
- [36]
-
A. Hagberg and E. Meron.
Pattern-formation in nongradient reaction-diffusion systems - the
effects of front bifurcations.
Nonlinearity, 7:805-835, 1994.
- [37]
-
Christian Elphick, Aric Hagberg, , and Ehud Meron.
Multi-phase patterns in periodically forced oscillatory systems.
Phys. Rev. E, 59:5285, 1999.
- [38]
-
D. Haim, G. Li, Q. Ouyang, W.D. McCormick, Harry L. Swinney, A. Hagberg, and
E. Meron.
Breathing spots in a reaction-diffusion system.
Phys. Rev. Lett., 77:190, 1996.
- [39]
-
C. Elphick, E. Meron, and E.A. Spiegel.
Patterns of propagating pulses.
SIAM J. Appl. Math, 50:490, 1990.
- [40]
-
C. Elphick, E. Meron, and E.A. Spiegel.
Spatiotemporal complexity in traveling patterns.
Phys. Rev. Lett., 61:496, 1988.
- [41]
-
N.J. Balmforth, G.R. Ierley, and E.A. Spiegel.
Chaotic pulse trains.
SIAM J. Appl. Math., 54:1291-1334, 1994.
- [42]
-
E. Knobloch.
Pattern selection in long-wavelength convection.
Physica D, 41:450, 1990.
File translated from
TEX
by
TTH,
version 2.60.
On 12 Mar 2000, 15:37.