Methods of Nonlinear Analysis D-12
Winter 2000
Hermann Riecke
Suggestions for Projects



  1. The impact of hidden symmetries on bifurcations [1,2,3]

  2. O(2)-Hopf bifurcation with weakly broken translation symmetry [4,5,6,7]

  3. Dynamics of dislocations in steady patterns within two-dimensional Ginzburg-Landau equation [8,9,10].

  4. The origin and justification of order-parameter models of the Swift-Hohenberg type. [11,12]

  5. 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]

  6. Localized solutions of the CGL and their role in spatio-temporally chaotic solutions [19,20,21,22,23,24,25]

  7. Bounded localized solutions of the 1-d CGL without saturation [26,27,28,29]

  8. Coupling of bifurcating modes to weakly damped modes: mean flow in Rayleigh-Bénard convection, concentration field in binary-mixture convection. [30,31,32]

  9. Front propagation into unstable states: velocity selection and the marginal stability criterion. [33,34,35]

  10. Instabilities of propagating fronts connecting two stable states, transition from stationary to traveling fronts, spot replication. [36,37,38]

  11. Interaction of localized structures, irregular sequences of pulses [39,40,41]

  12. 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.


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